ITP001 Axioms: ITP074^5.ax
%------------------------------------------------------------------------------
% File : ITP074^5 : 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 : ring^2.ax [Gau20]
% : HL4074^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 80 ( 0 unt; 16 typ; 0 def)
% Number of atoms : 2906 ( 143 equ; 0 cnn)
% Maximal formula atoms : 211 ( 36 avg)
% Number of connectives : 6835 ( 0 ~; 0 |; 117 &;6278 @)
% ( 7 <=>; 433 =>; 0 <=; 0 <~>)
% Maximal formula depth : 73 ( 16 avg;6278 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 17 ( 17 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 48 usr; 32 con; 0-2 aty)
% Number of variables : 524 ( 14 ^ 487 !; 23 ?; 524 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ering_2Ering,type,
ty_2Ering_2Ering: del > del ).
thf(tp_c_2Ering_2Eis__ring,type,
c_2Ering_2Eis__ring: del > $i ).
thf(mem_c_2Ering_2Eis__ring,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Eis__ring @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ bool ) ) ).
thf(tp_c_2Ering_2Erecordtype_2Ering,type,
c_2Ering_2Erecordtype_2Ering: del > $i ).
thf(mem_c_2Ering_2Erecordtype_2Ering,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2Ering_2Ering__CASE,type,
c_2Ering_2Ering__CASE: del > del > $i ).
thf(mem_c_2Ering_2Ering__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ering_2Ering__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) @ A_27b ) ) ) ).
thf(tp_c_2Ering_2Ering__R0,type,
c_2Ering_2Ering__R0: del > $i ).
thf(mem_c_2Ering_2Ering__R0,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Ering_2Ering__R0__fupd,type,
c_2Ering_2Ering__R0__fupd: del > $i ).
thf(mem_c_2Ering_2Ering__R0__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__R1,type,
c_2Ering_2Ering__R1: del > $i ).
thf(mem_c_2Ering_2Ering__R1,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Ering_2Ering__R1__fupd,type,
c_2Ering_2Ering__R1__fupd: del > $i ).
thf(mem_c_2Ering_2Ering__R1__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__RM,type,
c_2Ering_2Ering__RM: del > $i ).
thf(mem_c_2Ering_2Ering__RM,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__RM__fupd,type,
c_2Ering_2Ering__RM__fupd: del > $i ).
thf(mem_c_2Ering_2Ering__RM__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__RN,type,
c_2Ering_2Ering__RN: del > $i ).
thf(mem_c_2Ering_2Ering__RN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ) ).
thf(tp_c_2Ering_2Ering__RN__fupd,type,
c_2Ering_2Ering__RN__fupd: del > $i ).
thf(mem_c_2Ering_2Ering__RN__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__RP,type,
c_2Ering_2Ering__RP: del > $i ).
thf(mem_c_2Ering_2Ering__RP,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__RP__fupd,type,
c_2Ering_2Ering__RP__fupd: del > $i ).
thf(mem_c_2Ering_2Ering__RP__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ).
thf(tp_c_2Ering_2Ering__size,type,
c_2Ering_2Ering__size: del > $i ).
thf(mem_c_2Ering_2Ering__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Ering__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ering_2Esemi__ring__of,type,
c_2Ering_2Esemi__ring__of: del > $i ).
thf(mem_c_2Ering_2Esemi__ring__of,axiom,
! [A_27a: del] : ( mem @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ).
thf(ax_thm_2Ering_2Ering__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ ( ty_2Ering_2Ering @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ bool )
@ ^ [V2_27ring_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V6a2: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V7a3: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ A_27a ) )
@ ( lam @ ( arr @ A_27a @ A_27a )
@ ^ [V8a4: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( ap
@ ( ap
@ ( ap
@ ( lam @ A_27a
@ ^ [V9a0: $i] :
( lam @ A_27a
@ ^ [V10a1: $i] :
( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V11a2: $i] :
( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V12a3: $i] :
( lam @ ( arr @ A_27a @ A_27a )
@ ^ [V13a4: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) @ V9a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) @ V10a1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) @ V11a2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) @ V12a3 ) @ V13a4 ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) )
@ V4a0 )
@ V5a1 )
@ V6a2 )
@ V7a3 )
@ V8a4 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27ring_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27ring_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Ering_2Ering__case__def,axiom,
! [A_27a: del,A_27b: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2a2: $i] :
( ( mem @ V2a2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3a3: $i] :
( ( mem @ V3a3 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4a4: $i] :
( ( mem @ V4a4 @ ( arr @ A_27a @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) @ V4a4 ) ) @ V5f )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V5f @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) @ V4a4 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__size__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ A_27a )
=> ! [V3a2: $i] :
( ( mem @ V3a2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4a3: $i] :
( ( mem @ V4a3 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5a4: $i] :
( ( mem @ V5a4 @ ( arr @ A_27a @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ering_2Ering__size @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a0 ) @ V2a1 ) @ V3a2 ) @ V4a3 ) @ V5a4 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V1a0 ) ) @ ( ap @ V0f @ V2a1 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__R0,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V0a ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__R1,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V1a0 ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RP,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V2f ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RM,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V3f0 ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RN,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V4f1 ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__accessors,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) @ V4f1 ) )
= V0a ) ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ A_27a )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V8f0: $i] :
( ( mem @ V8f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V5a ) @ V6a0 ) @ V7f ) @ V8f0 ) @ V9f1 ) )
= V6a0 ) ) ) ) ) )
& ! [V10a: $i] :
( ( mem @ V10a @ A_27a )
=> ! [V11a0: $i] :
( ( mem @ V11a0 @ A_27a )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V13f0: $i] :
( ( mem @ V13f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V14f1: $i] :
( ( mem @ V14f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V10a ) @ V11a0 ) @ V12f ) @ V13f0 ) @ V14f1 ) )
= V12f ) ) ) ) ) )
& ! [V15a: $i] :
( ( mem @ V15a @ A_27a )
=> ! [V16a0: $i] :
( ( mem @ V16a0 @ A_27a )
=> ! [V17f: $i] :
( ( mem @ V17f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V18f0: $i] :
( ( mem @ V18f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V19f1: $i] :
( ( mem @ V19f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V15a ) @ V16a0 ) @ V17f ) @ V18f0 ) @ V19f1 ) )
= V18f0 ) ) ) ) ) )
& ! [V20a: $i] :
( ( mem @ V20a @ A_27a )
=> ! [V21a0: $i] :
( ( mem @ V21a0 @ A_27a )
=> ! [V22f: $i] :
( ( mem @ V22f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V23f0: $i] :
( ( mem @ V23f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V24f1: $i] :
( ( mem @ V24f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V20a ) @ V21a0 ) @ V22f ) @ V23f0 ) @ V24f1 ) )
= V24f1 ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__R0__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ A_27a @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ ( ap @ V0f2 @ V1a ) ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__R1__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ A_27a @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ ( ap @ V0f2 @ V2a0 ) ) @ V3f ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RP__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ ( ap @ V0f2 @ V3f ) ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RM__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ ( ap @ V0f2 @ V4f0 ) ) @ V5f1 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Ering__RN__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ ( ap @ V0f2 @ V5f1 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__fn__updates,axiom,
! [A_27a: del] :
( ! [V0f2: $i] :
( ( mem @ V0f2 @ ( arr @ A_27a @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V0f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ ( ap @ V0f2 @ V1a ) ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) )
& ! [V6f2: $i] :
( ( mem @ V6f2 @ ( arr @ A_27a @ A_27a ) )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ! [V8a0: $i] :
( ( mem @ V8a0 @ A_27a )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V10f0: $i] :
( ( mem @ V10f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V11f1: $i] :
( ( mem @ V11f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V6f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V7a ) @ V8a0 ) @ V9f ) @ V10f0 ) @ V11f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V7a ) @ ( ap @ V6f2 @ V8a0 ) ) @ V9f ) @ V10f0 ) @ V11f1 ) ) ) ) ) ) ) )
& ! [V12f2: $i] :
( ( mem @ V12f2 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V13a: $i] :
( ( mem @ V13a @ A_27a )
=> ! [V14a0: $i] :
( ( mem @ V14a0 @ A_27a )
=> ! [V15f: $i] :
( ( mem @ V15f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V16f0: $i] :
( ( mem @ V16f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V17f1: $i] :
( ( mem @ V17f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V12f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V13a ) @ V14a0 ) @ V15f ) @ V16f0 ) @ V17f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V13a ) @ V14a0 ) @ ( ap @ V12f2 @ V15f ) ) @ V16f0 ) @ V17f1 ) ) ) ) ) ) ) )
& ! [V18f2: $i] :
( ( mem @ V18f2 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V19a: $i] :
( ( mem @ V19a @ A_27a )
=> ! [V20a0: $i] :
( ( mem @ V20a0 @ A_27a )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V22f0: $i] :
( ( mem @ V22f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V23f1: $i] :
( ( mem @ V23f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V18f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V19a ) @ V20a0 ) @ V21f ) @ V22f0 ) @ V23f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V19a ) @ V20a0 ) @ V21f ) @ ( ap @ V18f2 @ V22f0 ) ) @ V23f1 ) ) ) ) ) ) ) )
& ! [V24f2: $i] :
( ( mem @ V24f2 @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V25a: $i] :
( ( mem @ V25a @ A_27a )
=> ! [V26a0: $i] :
( ( mem @ V26a0 @ A_27a )
=> ! [V27f: $i] :
( ( mem @ V27f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V28f0: $i] :
( ( mem @ V28f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V29f1: $i] :
( ( mem @ V29f1 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V24f2 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V25a ) @ V26a0 ) @ V27f ) @ V28f0 ) @ V29f1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V25a ) @ V26a0 ) @ V27f ) @ V28f0 ) @ ( ap @ V24f2 @ V29f1 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__accfupds,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V1f ) @ V0r ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) ) )
& ! [V2r: $i] :
( ( mem @ V2r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V3f ) @ V2r ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V2r ) ) ) )
& ! [V4r: $i] :
( ( mem @ V4r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V5f ) @ V4r ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V4r ) ) ) )
& ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V7f ) @ V6r ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V6r ) ) ) )
& ! [V8r: $i] :
( ( mem @ V8r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V9f ) @ V8r ) )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V8r ) ) ) )
& ! [V10r: $i] :
( ( mem @ V10r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V11f ) @ V10r ) )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V10r ) ) ) )
& ! [V12r: $i] :
( ( mem @ V12r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V13f ) @ V12r ) )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V12r ) ) ) )
& ! [V14r: $i] :
( ( mem @ V14r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V15f: $i] :
( ( mem @ V15f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V15f ) @ V14r ) )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V14r ) ) ) )
& ! [V16r: $i] :
( ( mem @ V16r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V17f: $i] :
( ( mem @ V17f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V17f ) @ V16r ) )
= ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V16r ) ) ) )
& ! [V18r: $i] :
( ( mem @ V18r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V19f: $i] :
( ( mem @ V19f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V19f ) @ V18r ) )
= ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V18r ) ) ) )
& ! [V20r: $i] :
( ( mem @ V20r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V21f ) @ V20r ) )
= ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V20r ) ) ) )
& ! [V22r: $i] :
( ( mem @ V22r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V23f: $i] :
( ( mem @ V23f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V23f ) @ V22r ) )
= ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V22r ) ) ) )
& ! [V24r: $i] :
( ( mem @ V24r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V25f: $i] :
( ( mem @ V25f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V25f ) @ V24r ) )
= ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V24r ) ) ) )
& ! [V26r: $i] :
( ( mem @ V26r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V27f: $i] :
( ( mem @ V27f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V27f ) @ V26r ) )
= ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V26r ) ) ) )
& ! [V28r: $i] :
( ( mem @ V28r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V29f: $i] :
( ( mem @ V29f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V29f ) @ V28r ) )
= ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V28r ) ) ) )
& ! [V30r: $i] :
( ( mem @ V30r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V31f: $i] :
( ( mem @ V31f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V31f ) @ V30r ) )
= ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V30r ) ) ) )
& ! [V32r: $i] :
( ( mem @ V32r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V33f: $i] :
( ( mem @ V33f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V33f ) @ V32r ) )
= ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V32r ) ) ) )
& ! [V34r: $i] :
( ( mem @ V34r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V35f: $i] :
( ( mem @ V35f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V35f ) @ V34r ) )
= ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V34r ) ) ) )
& ! [V36r: $i] :
( ( mem @ V36r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V37f: $i] :
( ( mem @ V37f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V37f ) @ V36r ) )
= ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V36r ) ) ) )
& ! [V38r: $i] :
( ( mem @ V38r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V39f: $i] :
( ( mem @ V39f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V39f ) @ V38r ) )
= ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V38r ) ) ) )
& ! [V40r: $i] :
( ( mem @ V40r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V41f: $i] :
( ( mem @ V41f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V41f ) @ V40r ) )
= ( ap @ V41f @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V40r ) ) ) ) )
& ! [V42r: $i] :
( ( mem @ V42r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V43f: $i] :
( ( mem @ V43f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V43f ) @ V42r ) )
= ( ap @ V43f @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V42r ) ) ) ) )
& ! [V44r: $i] :
( ( mem @ V44r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V45f: $i] :
( ( mem @ V45f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V45f ) @ V44r ) )
= ( ap @ V45f @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V44r ) ) ) ) )
& ! [V46r: $i] :
( ( mem @ V46r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V47f: $i] :
( ( mem @ V47f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V47f ) @ V46r ) )
= ( ap @ V47f @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V46r ) ) ) ) )
& ! [V48r: $i] :
( ( mem @ V48r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V49f: $i] :
( ( mem @ V49f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V49f ) @ V48r ) )
= ( ap @ V49f @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V48r ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__fupdfupds,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V1g ) @ V0r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V2f ) @ V1g ) ) @ V0r ) ) ) ) )
& ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V5f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V4g ) @ V3r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V5f ) @ V4g ) ) @ V3r ) ) ) ) )
& ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V8f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V7g ) @ V6r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V8f ) @ V7g ) ) @ V6r ) ) ) ) )
& ! [V9r: $i] :
( ( mem @ V9r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V10g: $i] :
( ( mem @ V10g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V11f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V10g ) @ V9r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V11f ) @ V10g ) ) @ V9r ) ) ) ) )
& ! [V12r: $i] :
( ( mem @ V12r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V13g: $i] :
( ( mem @ V13g @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V14f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V13g ) @ V12r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V14f ) @ V13g ) ) @ V12r ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__fupdfupds__comp,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V0g ) )
= ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V1f ) @ V0g ) ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27a ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V4f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V3g ) ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V4f ) @ V3g ) ) ) @ V2h ) ) ) ) )
& ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ A_27a ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V6f ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V5g ) )
= ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V6f ) @ V5g ) ) ) ) )
& ! [V7h: $i] :
( ( mem @ V7h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V8g: $i] :
( ( mem @ V8g @ ( arr @ A_27a @ A_27a ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V9f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V8g ) ) @ V7h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V9f ) @ V8g ) ) ) @ V7h ) ) ) ) )
& ! [V10g: $i] :
( ( mem @ V10g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V11f ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V10g ) )
= ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V11f ) @ V10g ) ) ) ) )
& ! [V12h: $i] :
( ( mem @ V12h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V13g: $i] :
( ( mem @ V13g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V14f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V13g ) ) @ V12h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V14f ) @ V13g ) ) ) @ V12h ) ) ) ) )
& ! [V15g: $i] :
( ( mem @ V15g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V16f: $i] :
( ( mem @ V16f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V16f ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V15g ) )
= ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V16f ) @ V15g ) ) ) ) )
& ! [V17h: $i] :
( ( mem @ V17h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V18g: $i] :
( ( mem @ V18g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V19f: $i] :
( ( mem @ V19f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V19f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V18g ) ) @ V17h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V19f ) @ V18g ) ) ) @ V17h ) ) ) ) )
& ! [V20g: $i] :
( ( mem @ V20g @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V21f ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V20g ) )
= ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V21f ) @ V20g ) ) ) ) )
& ! [V22h: $i] :
( ( mem @ V22h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V23g: $i] :
( ( mem @ V23g @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V24f: $i] :
( ( mem @ V24f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V24f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V23g ) ) @ V22h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V24f ) @ V23g ) ) ) @ V22h ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__fupdcanon,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V1g ) @ V0r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V1g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V2f ) @ V0r ) ) ) ) ) )
& ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V5f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V4g ) @ V3r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V4g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V5f ) @ V3r ) ) ) ) ) )
& ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27a ) )
=> ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V8f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V7g ) @ V6r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V7g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V8f ) @ V6r ) ) ) ) ) )
& ! [V9r: $i] :
( ( mem @ V9r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V10g: $i] :
( ( mem @ V10g @ ( arr @ A_27a @ A_27a ) )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V11f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V10g ) @ V9r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V10g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V11f ) @ V9r ) ) ) ) ) )
& ! [V12r: $i] :
( ( mem @ V12r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V13g: $i] :
( ( mem @ V13g @ ( arr @ A_27a @ A_27a ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V14f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V13g ) @ V12r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V13g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V14f ) @ V12r ) ) ) ) ) )
& ! [V15r: $i] :
( ( mem @ V15r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V16g: $i] :
( ( mem @ V16g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V17f: $i] :
( ( mem @ V17f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V17f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V16g ) @ V15r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V16g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V17f ) @ V15r ) ) ) ) ) )
& ! [V18r: $i] :
( ( mem @ V18r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V19g: $i] :
( ( mem @ V19g @ ( arr @ A_27a @ A_27a ) )
=> ! [V20f: $i] :
( ( mem @ V20f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V20f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V19g ) @ V18r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V19g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V20f ) @ V18r ) ) ) ) ) )
& ! [V21r: $i] :
( ( mem @ V21r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V22g: $i] :
( ( mem @ V22g @ ( arr @ A_27a @ A_27a ) )
=> ! [V23f: $i] :
( ( mem @ V23f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V23f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V22g ) @ V21r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V22g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V23f ) @ V21r ) ) ) ) ) )
& ! [V24r: $i] :
( ( mem @ V24r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V25g: $i] :
( ( mem @ V25g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V26f: $i] :
( ( mem @ V26f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V26f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V25g ) @ V24r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V25g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V26f ) @ V24r ) ) ) ) ) )
& ! [V27r: $i] :
( ( mem @ V27r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V28g: $i] :
( ( mem @ V28g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V29f: $i] :
( ( mem @ V29f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V29f ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V28g ) @ V27r ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V28g ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V29f ) @ V27r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__fupdcanon__comp,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V0g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V0g ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V1f ) ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27a ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V4f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V3g ) ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V3g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V4f ) ) @ V2h ) ) ) ) ) )
& ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ A_27a ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V6f ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V5g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V5g ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V6f ) ) ) ) )
& ! [V7h: $i] :
( ( mem @ V7h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V8g: $i] :
( ( mem @ V8g @ ( arr @ A_27a @ A_27a ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V9f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V8g ) ) @ V7h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V8g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V9f ) ) @ V7h ) ) ) ) ) )
& ! [V10g: $i] :
( ( mem @ V10g @ ( arr @ A_27a @ A_27a ) )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V11f ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V10g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V10g ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V11f ) ) ) ) )
& ! [V12h: $i] :
( ( mem @ V12h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V13g: $i] :
( ( mem @ V13g @ ( arr @ A_27a @ A_27a ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V14f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V13g ) ) @ V12h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V13g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V14f ) ) @ V12h ) ) ) ) ) )
& ! [V15g: $i] :
( ( mem @ V15g @ ( arr @ A_27a @ A_27a ) )
=> ! [V16f: $i] :
( ( mem @ V16f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V16f ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V15g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V15g ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V16f ) ) ) ) )
& ! [V17h: $i] :
( ( mem @ V17h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V18g: $i] :
( ( mem @ V18g @ ( arr @ A_27a @ A_27a ) )
=> ! [V19f: $i] :
( ( mem @ V19f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V19f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V18g ) ) @ V17h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V18g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V19f ) ) @ V17h ) ) ) ) ) )
& ! [V20g: $i] :
( ( mem @ V20g @ ( arr @ A_27a @ A_27a ) )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V21f ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V20g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V20g ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V21f ) ) ) ) )
& ! [V22h: $i] :
( ( mem @ V22h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V23g: $i] :
( ( mem @ V23g @ ( arr @ A_27a @ A_27a ) )
=> ! [V24f: $i] :
( ( mem @ V24f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V24f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V23g ) ) @ V22h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V23g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V24f ) ) @ V22h ) ) ) ) ) )
& ! [V25g: $i] :
( ( mem @ V25g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V26f: $i] :
( ( mem @ V26f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V26f ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V25g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V25g ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V26f ) ) ) ) )
& ! [V27h: $i] :
( ( mem @ V27h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V28g: $i] :
( ( mem @ V28g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V29f: $i] :
( ( mem @ V29f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V29f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V28g ) ) @ V27h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V28g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V29f ) ) @ V27h ) ) ) ) ) )
& ! [V30g: $i] :
( ( mem @ V30g @ ( arr @ A_27a @ A_27a ) )
=> ! [V31f: $i] :
( ( mem @ V31f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V31f ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V30g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V30g ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V31f ) ) ) ) )
& ! [V32h: $i] :
( ( mem @ V32h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V33g: $i] :
( ( mem @ V33g @ ( arr @ A_27a @ A_27a ) )
=> ! [V34f: $i] :
( ( mem @ V34f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V34f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V33g ) ) @ V32h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ V33g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V34f ) ) @ V32h ) ) ) ) ) )
& ! [V35g: $i] :
( ( mem @ V35g @ ( arr @ A_27a @ A_27a ) )
=> ! [V36f: $i] :
( ( mem @ V36f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V36f ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V35g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V35g ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V36f ) ) ) ) )
& ! [V37h: $i] :
( ( mem @ V37h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V38g: $i] :
( ( mem @ V38g @ ( arr @ A_27a @ A_27a ) )
=> ! [V39f: $i] :
( ( mem @ V39f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V39f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V38g ) ) @ V37h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ V38g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V39f ) ) @ V37h ) ) ) ) ) )
& ! [V40g: $i] :
( ( mem @ V40g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V41f: $i] :
( ( mem @ V41f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V41f ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V40g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V40g ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V41f ) ) ) ) )
& ! [V42h: $i] :
( ( mem @ V42h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V43g: $i] :
( ( mem @ V43g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V44f: $i] :
( ( mem @ V44f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V44f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V43g ) ) @ V42h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ V43g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V44f ) ) @ V42h ) ) ) ) ) )
& ! [V45g: $i] :
( ( mem @ V45g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V46f: $i] :
( ( mem @ V46f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V46f ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V45g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V45g ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V46f ) ) ) ) )
& ! [V47h: $i] :
( ( mem @ V47h @ ( arr @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) ) )
=> ! [V48g: $i] :
( ( mem @ V48g @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V49f: $i] :
( ( mem @ V49f @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V49f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V48g ) ) @ V47h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ V48g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Ering_2Ering @ A_27a ) @ ( ty_2Ering_2Ering @ A_27a ) ) @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ V49f ) ) @ V47h ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__component__equality,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( V0r1 = V1r2 )
<=> ( ( ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r1 )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V1r2 ) )
& ( ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r1 )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V1r2 ) )
& ( ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r1 )
= ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V1r2 ) )
& ( ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r1 )
= ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V1r2 ) )
& ( ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V0r1 )
= ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V1r2 ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__updates__eq__literal,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V5f ) ) @ V0r ) ) ) ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V5f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__literal__nchotomy,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ? [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
& ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ? [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27a ) )
& ( V0r
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V5f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2EFORALL__ring,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ bool ) )
=> ( ! [V1r: $i] :
( ( mem @ V1r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( p @ ( ap @ V0P @ V1r ) ) )
<=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f0: $i] :
( ( mem @ V5f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V5f0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V6f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2EEXISTS__ring,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ bool ) )
=> ( ? [V1r: $i] :
( ( mem @ V1r @ ( ty_2Ering_2Ering @ A_27a ) )
& ( p @ ( ap @ V0P @ V1r ) ) )
<=> ? [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
& ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V5f0: $i] :
( ( mem @ V5f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27a ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V5f0 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V6f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__literal__11,axiom,
! [A_27a: del,V0a01: $i] :
( ( mem @ V0a01 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2f11: $i] :
( ( mem @ V2f11 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f01: $i] :
( ( mem @ V3f01 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f1: $i] :
( ( mem @ V4f1 @ ( arr @ A_27a @ A_27a ) )
=> ! [V5a02: $i] :
( ( mem @ V5a02 @ A_27a )
=> ! [V6a2: $i] :
( ( mem @ V6a2 @ A_27a )
=> ! [V7f12: $i] :
( ( mem @ V7f12 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V8f02: $i] :
( ( mem @ V8f02 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V9f2: $i] :
( ( mem @ V9f2 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V0a01 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a1 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V2f11 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f01 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V4f1 ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__R0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V5a02 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__R1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V6a2 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V7f12 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V8f02 ) ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) @ V9f2 ) ) @ ( c_2Ebool_2EARB @ ( ty_2Ering_2Ering @ A_27a ) ) ) ) ) ) ) )
<=> ( ( V0a01 = V5a02 )
& ( V1a1 = V6a2 )
& ( V2f11 = V7f12 )
& ( V3f01 = V8f02 )
& ( V4f1 = V9f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Edatatype__ring,axiom,
! [A_27a: del,V0record: $i] :
( ( mem @ V0record @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ) ) ) )
=> ! [V1ring: $i] :
( ( mem @ V1ring @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V2R0: $i] :
( ( mem @ V2R0 @ A_27a )
=> ! [V3R1: $i] :
( ( mem @ V3R1 @ A_27a )
=> ! [V4RP: $i] :
( ( mem @ V4RP @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5RM: $i] :
( ( mem @ V5RM @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V6RN: $i] :
( ( mem @ V6RN @ ( arr @ A_27a @ A_27a ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0record @ V1ring ) @ V2R0 ) @ V3R1 ) @ V4RP ) @ V5RM ) @ V6RN ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__11,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2a2: $i] :
( ( mem @ V2a2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3a3: $i] :
( ( mem @ V3a3 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4a4: $i] :
( ( mem @ V4a4 @ ( arr @ A_27a @ A_27a ) )
=> ! [V5a0_27: $i] :
( ( mem @ V5a0_27 @ A_27a )
=> ! [V6a1_27: $i] :
( ( mem @ V6a1_27 @ A_27a )
=> ! [V7a2_27: $i] :
( ( mem @ V7a2_27 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V8a3_27: $i] :
( ( mem @ V8a3_27 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V9a4_27: $i] :
( ( mem @ V9a4_27 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) @ V4a4 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V5a0_27 ) @ V6a1_27 ) @ V7a2_27 ) @ V8a3_27 ) @ V9a4_27 ) )
<=> ( ( V0a0 = V5a0_27 )
& ( V1a1 = V6a1_27 )
& ( V2a2 = V7a2_27 )
& ( V3a3 = V8a3_27 )
& ( V4a4 = V9a4_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__nchotomy,axiom,
! [A_27a: del,V0rr: $i] :
( ( mem @ V0rr @ ( ty_2Ering_2Ering @ A_27a ) )
=> ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
& ? [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
& ( V0rr
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) )
=> ? [V1fn: $i] :
( ( mem @ V1fn @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ A_27b ) )
& ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ A_27a )
=> ! [V4a2: $i] :
( ( mem @ V4a2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5a3: $i] :
( ( mem @ V5a3 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V6a4: $i] :
( ( mem @ V6a4 @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ V1fn @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V2a0 ) @ V3a1 ) @ V4a2 ) @ V5a3 ) @ V6a4 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0f @ V2a0 ) @ V3a1 ) @ V4a2 ) @ V5a3 ) @ V6a4 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ bool ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) @ V5f1 ) ) ) ) ) ) ) )
=> ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( p @ ( ap @ V0P @ V6r ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__case__cong,axiom,
! [A_27a: del,A_27b: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V2M_27: $i] :
( ( mem @ V2M_27 @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) )
=> ( ( ( V1M = V2M_27 )
& ! [V4a0: $i] :
( ( mem @ V4a0 @ A_27a )
=> ! [V5a1: $i] :
( ( mem @ V5a1 @ A_27a )
=> ! [V6a2: $i] :
( ( mem @ V6a2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V7a3: $i] :
( ( mem @ V7a3 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V8a4: $i] :
( ( mem @ V8a4 @ ( arr @ A_27a @ A_27a ) )
=> ( ( V2M_27
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 ) @ V8a4 ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V3f @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 ) @ V8a4 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0f_27 @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 ) @ V8a4 ) ) ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ering_2Ering__CASE @ A_27a @ A_27b ) @ V1M ) @ V3f )
= ( ap @ ( ap @ ( c_2Ering_2Ering__CASE @ A_27a @ A_27b ) @ V2M_27 ) @ V0f_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ A_27b ) ) ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Ering_2Ering__CASE @ A_27a @ A_27b ) @ V0x ) @ V1f )
= V2v )
<=> ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4a0: $i] :
( ( mem @ V4a0 @ A_27a )
& ? [V5f_27: $i] :
( ( mem @ V5f_27 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V6f0: $i] :
( ( mem @ V6f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ A_27a @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Erecordtype_2Ering @ A_27a ) @ V3a ) @ V4a0 ) @ V5f_27 ) @ V6f0 ) @ V7f1 ) )
& ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V1f @ V3a ) @ V4a0 ) @ V5f_27 ) @ V6f0 ) @ V7f1 )
= V2v ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ering_2Eis__ring__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
<=> ( ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V2m ) @ V1n ) ) ) )
& ! [V3n: $i] :
( ( mem @ V3n @ A_27a )
=> ! [V4m: $i] :
( ( mem @ V4m @ A_27a )
=> ! [V5p: $i] :
( ( mem @ V5p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V3n ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V4m ) @ V5p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V3n ) @ V4m ) ) @ V5p ) ) ) ) )
& ! [V6n: $i] :
( ( mem @ V6n @ A_27a )
=> ! [V7m: $i] :
( ( mem @ V7m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V6n ) @ V7m )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V7m ) @ V6n ) ) ) )
& ! [V8n: $i] :
( ( mem @ V8n @ A_27a )
=> ! [V9m: $i] :
( ( mem @ V9m @ A_27a )
=> ! [V10p: $i] :
( ( mem @ V10p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V8n ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V9m ) @ V10p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V8n ) @ V9m ) ) @ V10p ) ) ) ) )
& ! [V11n: $i] :
( ( mem @ V11n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) @ V11n )
= V11n ) )
& ! [V12n: $i] :
( ( mem @ V12n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ V12n )
= V12n ) )
& ! [V13n: $i] :
( ( mem @ V13n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V13n ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V0r ) @ V13n ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) )
& ! [V14n: $i] :
( ( mem @ V14n @ A_27a )
=> ! [V15m: $i] :
( ( mem @ V15m @ A_27a )
=> ! [V16p: $i] :
( ( mem @ V16p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V14n ) @ V15m ) ) @ V16p )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V14n ) @ V16p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V15m ) @ V16p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Eplus__sym,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V2m ) @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Eplus__assoc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V2m ) @ V3p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__sym,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V2m ) @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__assoc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V2m ) @ V3p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Eplus__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) @ V1n )
= V1n ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__one__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ V1n )
= V1n ) ) ) ) ).
thf(conj_thm_2Ering_2Eopp__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V0r ) @ V1n ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Ering_2Edistr__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ V3p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V2m ) @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ering_2Eplus__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) )
= V1n ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) @ V1n )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) ) ) ) ).
thf(ax_thm_2Ering_2Esemi__ring__of__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Ering_2Ering__is__semi__ring,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Ering_2Emult__one__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) )
= V1n ) ) ) ) ).
thf(conj_thm_2Ering_2Eneg__mult,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V0r ) @ V1a ) ) @ V2b )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1a ) @ V2b ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------