ITP001 Axioms: ITP088^7.ax
%------------------------------------------------------------------------------
% File : ITP088^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : quotient_pred_set.ax [Gau19]
% : HL4088^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 98 ( 14 unt; 43 typ; 0 def)
% Number of atoms : 310 ( 32 equ; 4 cnn)
% Maximal formula atoms : 24 ( 3 avg)
% Number of connectives : 1384 ( 4 ~; 2 |; 14 &;1303 @)
% ( 12 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 12 avg;1303 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 468 ( 468 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 42 usr; 2 con; 0-8 aty)
% Number of variables : 404 ( 20 ^ 331 !; 1 ?; 404 :)
% ( 52 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_23_23,type,
c_2Epair_2E_23_23:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) ).
thf(c_2Equotient__pair_2E_23_23_23,type,
c_2Equotient__pair_2E_23_23_23:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c > $o ) > ( A_27b > A_27d > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( A_27c > A_27b ) > A_27a > A_27d ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Equotient__pred__set_2EDELETER,type,
c_2Equotient__pred__set_2EDELETER:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Equotient__pred__set_2EDISJOINTR,type,
c_2Equotient__pred__set_2EDISJOINTR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Equotient__pred__set_2EFINITER,type,
c_2Equotient__pred__set_2EFINITER:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Equotient__pred__set_2EGSPECR,type,
c_2Equotient__pred__set_2EGSPECR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ $o ) ) > A_27b > $o ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Equotient__pred__set_2EIMAGER,type,
c_2Equotient__pred__set_2EIMAGER:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Equotient__pred__set_2EINSERTR,type,
c_2Equotient__pred__set_2EINSERTR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EPSUBSET,type,
c_2Epred__set_2EPSUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Equotient__pred__set_2EPSUBSETR,type,
c_2Equotient__pred__set_2EPSUBSETR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Equotient__pred__set_2ESUBSETR,type,
c_2Equotient__pred__set_2ESUBSETR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Equotient_2Erespects,type,
c_2Equotient_2Erespects:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Equotient__pred__set_2EGSPECR__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ $o ),V3v: A_27b] :
( ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2f @ V3v )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 )
@ ^ [V4x: A_27a] : ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ $o @ A_27b @ $o @ V1R2 @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epair_2E_2C @ A_27b @ $o @ V3v @ c_2Ebool_2ET ) @ ( V2f @ V4x ) ) ) ) ).
thf(thm_2Equotient__pred__set_2ESUBSETR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Equotient__pred__set_2ESUBSETR @ A_27a @ V0R @ V1s @ V2t )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EPSUBSETR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a @ V0R @ V1s @ V2t )
<=> ( ( c_2Equotient__pred__set_2ESUBSETR @ A_27a @ V0R @ V1s @ V2t )
& ( (~) @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V1s @ V2t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EINSERTR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2s: A_27a > $o] :
( ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V1x @ V2s )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3y @ ( c_2Ebool_2E_5C_2F @ ( V0R @ V3y @ V1x ) @ ( c_2Ebool_2EIN @ A_27a @ V3y @ V2s ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDELETER__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a] :
( ( c_2Equotient__pred__set_2EDELETER @ A_27a @ V0R @ V1s @ V2x )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3y @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) @ ( c_2Ebool_2E_7E @ ( V0R @ V2x @ V3y ) ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDISJOINTR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a @ V0R @ V1s @ V2t )
<=> ( (~)
@ ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V1s )
= ( c_2Ebool_2ERES__FORALL @ ( ( A_27a > $o ) > $o ) @ ( c_2Equotient_2Erespects @ ( ( A_27a > $o ) > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V2P: ( A_27a > $o ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_2F_5C @ ( V2P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
@ ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ ( c_2Equotient_2Erespects @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V3s: A_27a > $o] :
( c_2Emin_2E_3D_3D_3E @ ( V2P @ V3s )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V4e: A_27a] : ( V2P @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V4e @ V3s ) ) ) ) ) )
@ ( V2P @ V1s ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIMAGER__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f: A_27a > A_27b,V3s: A_27a > $o] :
( ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2f @ V3s )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V4y: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V4y
@ ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 )
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V1R2 @ V4y @ ( V2f @ V5x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V3s ) ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__SET__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V0f @ ( c_2Ecombin_2EI @ $o ) @ V1s ) )
= ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V2x ) @ V1s ) ) ).
thf(thm_2Equotient__pred__set_2ESET__REL,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V1s @ V2t )
<=> ! [V3x: A_27a,V4y: A_27a] :
( ( V0R @ V3x @ V4y )
=> ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
= ( c_2Ebool_2EIN @ A_27a @ V4y @ V2t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2ESET__REL__MP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27a > $o,V4t: A_27a > $o,V5x: A_27a,V6y: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s @ V4t )
& ( V0R @ V5x @ V6y ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V3s )
= ( c_2Ebool_2EIN @ A_27a @ V6y @ V4t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27b,V4s: A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27b @ V3x @ V4s )
= ( c_2Ebool_2EIN @ A_27a @ ( V2rep @ V3x ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4s ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x1: A_27a,V4x2: A_27a,V5s1: A_27a > $o,V6s2: A_27a > $o] :
( ( ( V0R @ V3x1 @ V4x2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5s1 @ V6s2 ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V3x1 @ V5s1 )
= ( c_2Ebool_2EIN @ A_27a @ V4x2 @ V6s2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__GSPECR,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27b > A_27b > $o,V1R2: A_27a > A_27a > $o,V2f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ),V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Equotient__pred__set_2EGSPECR @ A_27b @ A_27a @ V0R1 @ V1R2 @ V2f ) )
= ( c_2Ebool_2ERES__EXISTS @ A_27b @ ( c_2Equotient_2Erespects @ A_27b @ $o @ V0R1 )
@ ^ [V4x: A_27b] : ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ $o @ A_27a @ $o @ V1R2 @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epair_2E_2C @ A_27a @ $o @ V3v @ c_2Ebool_2ET ) @ ( V2f @ V4x ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EGSPEC__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > ( tyop_2Epair_2Eprod @ A_27d @ $o )] :
( ( c_2Epred__set_2EGSPEC @ A_27d @ A_27c @ V6f )
= ( c_2Equotient_2E_2D_2D_3E @ A_27d @ $o @ A_27b @ $o @ V5rep2 @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b @ V0R1 @ V3R2 @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( tyop_2Epair_2Eprod @ A_27d @ $o ) @ A_27c @ ( tyop_2Epair_2Eprod @ A_27b @ $o ) @ V1abs1 @ ( c_2Epair_2E_23_23 @ A_27d @ $o @ A_27b @ $o @ V5rep2 @ ( c_2Ecombin_2EI @ $o ) ) @ V6f ) ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EGSPECR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f1: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ $o ),V7f2: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ $o )] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ $o ) @ V0R1 @ ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ $o @ A_27b @ $o @ V3R2 @ ( c_2Emin_2E_3D @ $o ) ) @ V6f1 @ V7f2 )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ $o @ V3R2 @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f1 ) @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b @ V0R1 @ V3R2 @ V7f2 ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EEMPTY__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( ( c_2Epred__set_2EEMPTY @ A_27b )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EEMPTY__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Equotient__pred__set_2EUNIV__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( ( c_2Epred__set_2EUNIV @ A_27b )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EUNIV__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Equotient__pred__set_2EUNION__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EUNION @ A_27b @ V3s @ V4t )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EUNION__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V3s1 @ V5t1 ) @ ( c_2Epred__set_2EUNION @ A_27a @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EINTER__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EINTER @ A_27b @ V3s @ V4t )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EINTER__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V3s1 @ V5t1 ) @ ( c_2Epred__set_2EINTER @ A_27a @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2ESUBSET__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V3s @ V4t )
= ( c_2Equotient__pred__set_2ESUBSETR @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2ESUBSETR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( ( c_2Equotient__pred__set_2ESUBSETR @ A_27a @ V0R @ V3s1 @ V5t1 )
= ( c_2Equotient__pred__set_2ESUBSETR @ A_27a @ V0R @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EPSUBSET__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27b @ V3s @ V4t )
= ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EPSUBSETR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a @ V0R @ V3s1 @ V5t1 )
= ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a @ V0R @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__INSERTR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2s: A_27a > $o,V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V1x @ V2s ) )
<=> ( ( V0R @ V3y @ V1x )
| ( c_2Ebool_2EIN @ A_27a @ V3y @ V2s ) ) ) ).
thf(thm_2Equotient__pred__set_2EINSERT__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4x: A_27b] :
( ( c_2Epred__set_2EINSERT @ A_27b @ V4x @ V3s )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ ( V2rep @ V4x ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EINSERTR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x1: A_27a,V4x2: A_27a,V5s1: A_27a > $o,V6s2: A_27a > $o] :
( ( ( V0R @ V3x1 @ V4x2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5s1 @ V6s2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V3x1 @ V5s1 ) @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V4x2 @ V6s2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__DELETER,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Equotient__pred__set_2EDELETER @ A_27a @ V0R @ V1s @ V2x ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( (~) @ ( V0R @ V2x @ V3y ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDELETE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4x: A_27b] :
( ( c_2Epred__set_2EDELETE @ A_27b @ V3s @ V4x )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Equotient__pred__set_2EDELETER @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( V2rep @ V4x ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDELETER__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5x1: A_27a,V6x2: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( V0R @ V5x1 @ V6x2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Equotient__pred__set_2EDELETER @ A_27a @ V0R @ V3s1 @ V5x1 ) @ ( c_2Equotient__pred__set_2EDELETER @ A_27a @ V0R @ V4s2 @ V6x2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDIFF__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EDIFF @ A_27b @ V3s @ V4t )
= ( c_2Equotient_2E_2D_2D_3E @ A_27b @ $o @ A_27a @ $o @ V2rep @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDIFF__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V3s1 @ V5t1 ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDISJOINT__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27b @ V3s @ V4t )
= ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4t ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EDISJOINTR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o,V5t1: A_27a > $o,V6t2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5t1 @ V6t2 ) )
=> ( ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a @ V0R @ V3s1 @ V5t1 )
= ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a @ V0R @ V4s2 @ V6t2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
= ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3s ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__EQ,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s1: A_27a > $o,V2s2: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V1s1 @ V2s2 )
=> ( ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V1s1 )
= ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V2s2 ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3s1: A_27a > $o,V4s2: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3s1 @ V4s2 )
=> ( ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V3s1 )
= ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V4s2 ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__EMPTY,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__INSERTR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ ( c_2Equotient_2Erespects @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V1s: A_27a > $o] :
( c_2Emin_2E_3D_3D_3E @ ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V1s )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V2x: A_27a] : ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V2x @ V1s ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EABSORPTIONR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V1x: A_27a] :
( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ ( c_2Equotient_2Erespects @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V2s: A_27a > $o] : ( c_2Emin_2E_3D @ $o @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2s ) @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V1x @ V2s ) @ V2s ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EFINITER__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( c_2Ebool_2ERES__FORALL @ ( ( A_27a > $o ) > $o ) @ ( c_2Equotient_2Erespects @ ( ( A_27a > $o ) > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V1P: ( A_27a > $o ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_2F_5C @ ( V1P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
@ ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ ( c_2Equotient_2Erespects @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V2s: A_27a > $o] :
( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V2s ) @ ( V1P @ V2s ) )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V3e: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V3e @ V2s ) ) @ ( V1P @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a @ V0R @ V3e @ V2s ) ) ) ) ) ) )
@ ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ ( c_2Equotient_2Erespects @ ( A_27a > $o ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) ) )
@ ^ [V4s: A_27a > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Equotient__pred__set_2EFINITER @ A_27a @ V0R @ V4s ) @ ( V1P @ V4s ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIN__IMAGER,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2y: A_27b,V3f: A_27a > A_27b,V4s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27b @ V2y @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b @ V0R1 @ V1R2 @ V3f @ V4s ) )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 )
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V1R2 @ V2y @ ( V3f @ V5x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V4s ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIMAGE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d,V7s: A_27c > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27c @ A_27d @ V6f @ V7s )
= ( c_2Equotient_2E_2D_2D_3E @ A_27d @ $o @ A_27b @ $o @ V5rep2 @ ( c_2Ecombin_2EI @ $o ) @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b @ V0R1 @ V3R2 @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27c @ $o @ V1abs1 @ ( c_2Ecombin_2EI @ $o ) @ V7s ) ) ) ) ) ) ).
thf(thm_2Equotient__pred__set_2EIMAGER__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f1: A_27a > A_27b,V7f2: A_27a > A_27b,V8s1: A_27a > $o,V9s2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f1 @ V7f2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R1 @ ( c_2Emin_2E_3D @ $o ) @ V8s1 @ V9s2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ $o @ V3R2 @ ( c_2Emin_2E_3D @ $o ) @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f1 @ V8s1 ) @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b @ V0R1 @ V3R2 @ V7f2 @ V9s2 ) ) ) ) ) ).
%------------------------------------------------------------------------------