ITP001 Axioms: ITP138^5.ax
%------------------------------------------------------------------------------
% File : ITP138^5 : 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 : real_sigma^2.ax [Gau20]
% : HL4138^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 0 unt; 1 typ; 0 def)
% Number of atoms : 1131 ( 39 equ; 0 cnn)
% Maximal formula atoms : 41 ( 31 avg)
% Number of connectives : 1782 ( 6 ~; 0 |; 20 &;1607 @)
% ( 1 <=>; 148 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 14 avg;1607 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 55 usr; 54 con; 0-2 aty)
% Number of variables : 163 ( 26 ^ 136 !; 1 ?; 163 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ereal__sigma_2EREAL__SUM__IMAGE,type,
c_2Ereal__sigma_2EREAL__SUM__IMAGE: del > $i ).
thf(mem_c_2Ereal__sigma_2EREAL__SUM__IMAGE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(ax_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DEF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ ty_2Erealax_2Ereal )
@ ( lam @ A_27a
@ ^ [V2e: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3acc: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V2e ) ) @ V3acc ) ) ) )
@ V1s )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__THM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V1e ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SPOS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NONZERO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0P
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IF__ELIM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ V1P @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2p ) @ V0P ) )
& ! [V3q: $i] :
( ( mem @ V3q @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3q ) @ V0P ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V2p ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3q ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) @ ( ap @ V1f @ V2p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3y ) )
= V2x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) ) @ ( ap @ V1f @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CMUL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1f @ V3x ) ) ) )
@ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NEG,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1f @ V2x ) ) ) )
@ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V2f ) @ V1f_27 ) ) @ V0P ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1P_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0P ) @ V1P_27 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0P ) @ V1P_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V1P_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__CARD,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ( V0P
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V1s: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1s ) @ V0P ) ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f )
@ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P )
@ ( lam @ A_27a
@ ^ [V2p: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V2p ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2P_27: $i] :
( ( mem @ V2P_27 @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2P_27 ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P ) @ V2P_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__COUNT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0P ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CONST__EQ__1__EQ__INV__CARD,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0P ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V2x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3y ) ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0P ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V4x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__ADD,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) )
@ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__REAL__SUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s_27 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1s_27 ) ) )
@ V0s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4x: $i] : ( ap @ ( ap @ V2f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V4x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2ESEQ__REAL__SUM__IMAGE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ V1f @ V4n ) @ V3x ) ) )
@ ( ap @ V2f_27 @ V3x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( ap @ V1f @ V5n ) ) @ V0s ) ) )
@ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2ENESTED__REAL__SUM__IMAGE__REVERSE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s_27 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2s_27 ) ) )
@ V1s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V4x: $i] :
( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V5y: $i] : ( ap @ ( ap @ V0f @ V5y ) @ V4x ) ) )
@ V1s ) ) )
@ V2s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__sum,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1r ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum ) @ V1r ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POW,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0a ) @ V1s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ty_2Erealax_2Ereal )
@ ( lam @ A_27a
@ ^ [V2i: $i] :
( lam @ A_27a
@ ^ [V3j: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V2i ) ) @ ( ap @ V0a @ V3j ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V1s ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f_27 @ V3x ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
@ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SUB,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) )
@ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO__SET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ty_2Erealax_2Ereal ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s2 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ty_2Erealax_2Ereal )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V1s1 ) @ V2s2 ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27a @ ty_2Erealax_2Ereal )
@ ( lam @ A_27b
@ ^ [V5y: $i] :
( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V6x ) @ V5y ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a ) @ V2s2 ) @ V1s1 ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------