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 : 78 ( 0 unt; 4 typ; 0 def)
% Number of atoms : 2953 ( 39 equ; 0 cnn)
% Maximal formula atoms : 83 ( 37 avg)
% Number of connectives : 3934 ( 13 ~; 0 |; 137 &;3586 @)
% ( 11 <=>; 187 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 17 avg;3586 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 45 con; 0-2 aty)
% Number of variables : 350 ( 38 ^ 285 !; 27 ?; 350 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Elim_2Econtl,type,
c_2Elim_2Econtl: $i ).
thf(mem_c_2Elim_2Econtl,axiom,
mem @ c_2Elim_2Econtl @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Elim_2Edifferentiable,type,
c_2Elim_2Edifferentiable: $i ).
thf(mem_c_2Elim_2Edifferentiable,axiom,
mem @ c_2Elim_2Edifferentiable @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Elim_2Ediffl,type,
c_2Elim_2Ediffl: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Elim_2Etends__real__real,type,
c_2Elim_2Etends__real__real: $i ).
thf(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 ) ) ) ).
thf(ax_thm_2Elim_2Etends__real__real,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ 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 ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1y0: tp__ty_2Erealax_2Ereal,V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1y0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
<=> ! [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 ) ) )
=> ? [V4d: 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 @ V4d ) ) )
& ! [V5x: 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 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y0 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__CONST,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V0k ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ).
thf(conj_thm_2Elim_2ELIM__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__MUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( V1l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__DIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( V3m
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__NULL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__X,axiom,
! [V0x0: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ ( inj__ty_2Erealax_2Ereal @ V0x0 ) )
@ ( inj__ty_2Erealax_2Ereal @ V0x0 ) ) ) ).
thf(conj_thm_2Elim_2ELIM__UNIQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2m: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ( V1l = V2m ) ) ) ).
thf(conj_thm_2Elim_2ELIM__EQUAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3x0: tp__ty_2Erealax_2Ereal] :
( ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( V4x != V3x0 )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x0 ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ELIM__TRANSFORM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) ) ) ) ).
thf(ax_thm_2Elim_2Ediffl,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3h: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3h ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ V3h ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(ax_thm_2Elim_2Econtl,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2h: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2h ) ) ) )
@ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(ax_thm_2Elim_2Edifferentiable,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Elim_2Edifferentiable @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
<=> ? [V2l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__UNIQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2m: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ( V1l = V2m ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__CONT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONTL__LIM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__CARAT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
<=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ! [V4z: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= V1l ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__CONST,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ).
thf(conj_thm_2Elim_2ECONT__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__MUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V2x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__DIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Elim_2Econtl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ V1g @ ( ap @ V0f @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EIVT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ? [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V3y ) ) ) ) ).
thf(conj_thm_2Elim_2EIVT2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ? [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V3y ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__CONST,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__MUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V4x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__CHAIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ V0f @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__X,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__POW,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ V2x ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__XM1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( V0x
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Einv @ V1x ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__DIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__SUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum,V4x: tp__ty_2Erealax_2Ereal] :
( ! [V5r: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V5r ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5r ) ) @ V6x ) ) )
@ ( ap @ ( ap @ V1f_27 @ ( inj__ty_2Enum_2Enum @ V5r ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] :
( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ ( ap @ V0f @ V8n ) @ V7x ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9r: $i] : ( ap @ ( ap @ V1f_27 @ V9r ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__HASSUP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) ) )
& ! [V6N: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6N ) ) @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) )
=> ? [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6N ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__ATTAINS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) ) )
& ? [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V4M ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__ATTAINS2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
& ? [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V4M ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__ATTAINS__ALL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4L: tp__ty_2Erealax_2Ereal,V5M: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4L ) ) @ ( inj__ty_2Erealax_2Ereal @ V5M ) ) )
& ! [V6y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4L ) ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ ( inj__ty_2Erealax_2Ereal @ V5M ) ) ) )
=> ? [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= V6y ) ) )
& ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4L ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5M ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__LINC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( 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 @ V2l ) ) ) )
=> ? [V3d: 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 @ V3d ) ) )
& ! [V4h: 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 @ V4h ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4h ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4h ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__LDEC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V3d: 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 @ V3d ) ) )
& ! [V4h: 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 @ V4h ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4h ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4h ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__LMAX,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ? [V3d: 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 @ V3d ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) )
=> ( V2l
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__LMIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ? [V3d: 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 @ V3d ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) )
=> ( V2l
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__LCONST,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ? [V3d: 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 @ V3d ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) )
=> ( V2l
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EINTERVAL__LEMMA,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ? [V3d: 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 @ V3d ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EROLLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Edifferentiable @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ? [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EMVT__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( 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 @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ V3x ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( 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 @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ V4x ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ).
thf(conj_thm_2Elim_2EMVT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Edifferentiable @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ? [V5l: tp__ty_2Erealax_2Ereal,V6z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__ISCONST__END,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__ISCONST,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__ISCONST__ALL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EINTERVAL__ABS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1z: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__INJ__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: 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 @ V3d ) ) )
& ! [V4z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) )
= V4z ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) )
=> ~ ! [V6z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__INJ__LEMMA2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: 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 @ V3d ) ) )
& ! [V4z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) )
= V4z ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) )
=> ~ ! [V6z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__INJ__RANGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: 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 @ V3d ) ) )
& ! [V4z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) )
= V4z ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) )
=> ? [V6e: 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 @ V6e ) ) )
& ! [V7y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) )
=> ? [V8z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V8z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8z ) ) )
= V7y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2ECONT__INVERSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: 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 @ V3d ) ) )
& ! [V4z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) )
= V4z ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V1g ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__INVERSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal,V4d: 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 @ V4d ) ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) )
= V5z ) )
& ! [V6z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( V2l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__INVERSE__LT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal,V4d: 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 @ V4d ) ) )
& ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) )
= V5z ) )
& ! [V6z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( V2l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EINTERVAL__CLEMMA,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ? [V3d: 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 @ V3d ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elim_2EDIFF__INVERSE__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3a: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
& ! [V6z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
= V6z )
& ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( V2l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------