ITP001 Axioms: ITP147^5.ax
%------------------------------------------------------------------------------
% File : ITP147^5 : 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 : product^2.ax [Gau20]
% : HL4147^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 7 unt; 2 typ; 0 def)
% Number of atoms : 3359 ( 101 equ; 0 cnn)
% Maximal formula atoms : 72 ( 38 avg)
% Number of connectives : 4575 ( 2 ~; 0 |; 63 &;4253 @)
% ( 4 <=>; 253 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg;4253 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 60 usr; 58 con; 0-2 aty)
% Number of variables : 365 ( 38 ^ 323 !; 4 ?; 365 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eproduct_2Enproduct,type,
c_2Eproduct_2Enproduct: del > $i ).
thf(mem_c_2Eproduct_2Enproduct,axiom,
! [A_27a: del] : ( mem @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Eproduct_2Eproduct,type,
c_2Eproduct_2Eproduct: del > $i ).
thf(mem_c_2Eproduct_2Eproduct,axiom,
! [A_27a: del] : ( mem @ ( c_2Eproduct_2Eproduct @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Eproduct_2EREAL__ADD__AC,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Erealax_2Ereal,V2p: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2p ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( inj__ty_2Erealax_2Ereal @ V2p ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( inj__ty_2Erealax_2Ereal @ V2p ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EMULT__AC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) ) ).
thf(ax_thm_2Eproduct_2Enproduct,axiom,
! [A_27a: del] :
( ( c_2Eproduct_2Enproduct @ A_27a )
= ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2A ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1x ) @ V3s ) ) @ V2f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27b ) @ V3s ) @ V2f ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V2f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27b ) @ V3s ) @ V2f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__SUPPORT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2A ) @ V0f ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__UNION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [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_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V1g ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__ADD__SPLIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__POS__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [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_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__OFFSET,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V1f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4i ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__SING__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) )
= fo__c_2Enum_2E0 )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V2x ) )
= fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= fo__c_2Enum_2E0 )
<=> ? [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__LE,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1f @ V3x ) ) @ ( ap @ V0g @ V3x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V0g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__LE__NUMSEG,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ__1,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V2x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__EQ__1__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__MUL__GEN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V1g @ V4x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__MUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__MUL__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CONST,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3x: $i] : ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__ONE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1n: $i] : ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V3x: tp__ty_2Enum_2Enum,V4y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5a ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ V5a ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2s ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V2v ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V2a ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2a ) ) @ V0f ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__FACT,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : V1m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eproduct_2ENPRODUCT__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eproduct_2Eproduct,axiom,
! [A_27a: del] :
( ( c_2Eproduct_2Eproduct @ A_27a )
= ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__mul ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f ) )
= ( 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 ) ) ) ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1x ) @ V3s ) ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b ) @ V3s ) @ V2f ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b ) @ V3s ) @ V2f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__SUPPORT,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_2Eproduct_2Eproduct @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__mul ) @ V0f ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__UNION,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_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V1g ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__ADD__SPLIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__POS__LE,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_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__POS__LE__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__POS__LT,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_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ 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_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__OFFSET,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V1f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4i ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__SING__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__EQ__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__EQ__0,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 ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V2x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ? [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__LE,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ V0g @ V3x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V0g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__LE__NUMSEG,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__EQ__1,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V2x ) )
= ( 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 ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) )
= ( 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_2Eproduct_2EPRODUCT__EQ__1__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
= ( 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 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( 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_2Eproduct_2EPRODUCT__MUL__GEN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V4x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__MUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__MUL__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CONST,axiom,
! [A_27a: del,V0c: tp__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_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__NEG,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 ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2i: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__NEG__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__NEG__NUMSEG__1,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__INV,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 ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V2x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__DIV,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__DIV__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__ONE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( 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_2Eproduct_2EPRODUCT__LE__1,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 @ V0f @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) @ ( 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_2Eproduct_2EPRODUCT__ABS,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 ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5a ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ V5a ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EREAL__OF__NUM__NPRODUCT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ ( c_2Eproduct_2Enproduct @ A_27a ) @ V1s ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ V0f @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( 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 ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2v ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V2a ) ) @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2a ) ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eproduct_2EPRODUCT__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) @ ( 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_2Eproduct_2Eth,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V0f @ V4i ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7a: tp__ty_2Enum_2Enum,V8b: tp__ty_2Enum_2Enum] :
( ! [V9i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V9i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V9i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V6g @ ( inj__ty_2Enum_2Enum @ V9i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10i: $i] : ( ap @ V5f @ V10i ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) ) @ V6g ) ) ) ) ) )
& ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V12g: $i] :
( ( mem @ V12g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V13p: $i] :
( ( mem @ V13p @ ( arr @ A_27b @ bool ) )
=> ( ! [V14x: $i] :
( ( mem @ V14x @ A_27b )
=> ( ( p @ ( ap @ V13p @ V14x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V11f @ V14x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V12g @ V14x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V15y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V15y ) @ ( ap @ V13p @ V15y ) ) ) ) )
@ ( lam @ A_27b
@ ^ [V16i: $i] : ( ap @ V11f @ V16i ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V17y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V17y ) @ ( ap @ V13p @ V17y ) ) ) ) )
@ V12g ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------