SET007 Axioms: SET007+75.ax
%------------------------------------------------------------------------------
% File : SET007+75 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Consequences of the Reflection Theorem
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : zfrefle1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 5 unt; 0 def)
% Number of atoms : 378 ( 46 equ)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 413 ( 91 ~; 9 |; 141 &)
% ( 14 <=>; 158 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-3 aty)
% Number of functors : 48 ( 48 usr; 17 con; 0-4 aty)
% Number of variables : 142 ( 139 !; 3 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k9_zf_lang))
=> ( r1_zfrefle1(A,B)
<=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_hidden(C,B)
=> r2_zf_model(A,C) ) ) ) ) ) ).
fof(d2_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r2_zfrefle1(A,B)
<=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( k2_zf_model(C) = k1_xboole_0
=> ( r2_zf_model(A,C)
<=> r2_zf_model(B,C) ) ) ) ) ) ) ).
fof(d3_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r3_zfrefle1(A,B)
<=> ( r1_tarski(A,B)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( r1_zf_model(A,D,C)
<=> r1_zf_model(B,k2_zf_lang1(k1_zf_lang,A,B,D),C) ) ) ) ) ) ) ) ).
fof(d4_zfrefle1,axiom,
! [A] :
( A = k1_zfrefle1
<=> ! [B] :
( r2_hidden(B,A)
<=> ( r2_hidden(B,k9_zf_lang)
& ~ ( B != k6_zf_model
& B != k7_zf_model
& B != k8_zf_model
& B != k9_zf_model
& B != k10_zf_model
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(C))
& B = k11_zf_model(C) ) ) ) ) ) ) ).
fof(t1_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r1_zfrefle1(A,k1_subset_1(k9_zf_lang)) ) ).
fof(t2_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k9_zf_lang))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k9_zf_lang))
=> ( ( r1_tarski(B,C)
& r1_zfrefle1(A,C) )
=> r1_zfrefle1(A,B) ) ) ) ) ).
fof(t3_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k9_zf_lang))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k9_zf_lang))
=> ( ( r1_zfrefle1(A,B)
& r1_zfrefle1(A,C) )
=> r1_zfrefle1(A,k4_subset_1(k9_zf_lang,B,C)) ) ) ) ) ).
fof(t4_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_zf_model(A)
=> r1_zfrefle1(A,k2_zfrefle1) ) ) ).
fof(t5_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ( r1_zfrefle1(A,k2_zfrefle1)
& v1_ordinal1(A) )
=> v1_zf_model(A) ) ) ).
fof(t6_zfrefle1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& k2_zf_model(B) = k1_xboole_0
& ! [C] :
( ~ v1_xboole_0(C)
=> ( r2_zf_model(C,B)
<=> r2_zf_model(C,A) ) ) ) ) ).
fof(t7_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r2_zfrefle1(A,B)
<=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_zf_model(A,C)
<=> r2_zf_model(B,C) ) ) ) ) ) ).
fof(t8_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r2_zfrefle1(A,B)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k9_zf_lang))
=> ( r1_zfrefle1(A,C)
<=> r1_zfrefle1(B,C) ) ) ) ) ) ).
fof(t9_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r3_zfrefle1(A,B)
=> r2_zfrefle1(A,B) ) ) ) ).
fof(t10_zfrefle1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( ( v1_zf_model(A)
& r2_zfrefle1(A,B)
& v1_ordinal1(B) )
=> v1_zf_model(B) ) ) ) ).
fof(t11_zfrefle1,axiom,
$true ).
fof(t12_zfrefle1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( ( r2_hidden(k1_relat_1(A),B)
& r1_tarski(k2_relat_1(A),B) )
=> r2_hidden(k2_relat_1(A),B) ) ) ) ).
fof(t13_zfrefle1,axiom,
! [A,B] :
( ( r2_wellord2(A,B)
| k1_card_1(A) = k1_card_1(B) )
=> ( r2_wellord2(k1_zfmisc_1(A),k1_zfmisc_1(B))
& k1_card_1(k1_zfmisc_1(A)) = k1_card_1(k1_zfmisc_1(B)) ) ) ).
fof(t14_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_funct_2(k2_ordinal2(A),k2_ordinal2(A)))
& m2_relset_1(C,B,k1_funct_2(k2_ordinal2(A),k2_ordinal2(A))) )
=> ~ ( r2_hidden(k1_card_1(B),k1_card_1(A))
& ! [D] :
( m2_ordinal4(D,A)
=> ~ ( v2_ordinal2(D)
& v3_ordinal2(D)
& k4_ordinal4(A,D,k2_ordinal4(A)) = k2_ordinal4(A)
& ! [E] :
( m1_ordinal4(E,A)
=> k4_ordinal4(A,D,k6_ordinal4(A,E)) = k7_ordinal2(k2_xboole_0(k1_tarski(k4_ordinal4(A,D,E)),k9_relat_1(k4_funct_5(C),k2_zfmisc_1(B,k1_tarski(k6_ordinal4(A,E)))))) )
& ! [E] :
( m1_ordinal4(E,A)
=> ( v4_ordinal1(E)
=> ( E = k2_ordinal4(A)
| k4_ordinal4(A,D,E) = k8_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) ) ).
fof(t15_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v2_ordinal2(B)
=> v2_ordinal2(k1_ordinal3(A,B)) ) ) ) ).
fof(t16_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> k2_ordinal1(k1_ordinal3(A,C),B) = k1_ordinal3(A,k2_ordinal1(C,B)) ) ) ) ).
fof(t17_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( ( v2_ordinal2(B)
& v3_ordinal2(B) )
=> v3_ordinal2(k1_ordinal3(A,B)) ) ) ) ).
fof(d5_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r4_zfrefle1(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& k1_relat_1(C) = B
& r1_tarski(k2_relat_1(C),A)
& v2_ordinal2(C)
& A = k8_ordinal2(C) ) ) ) ) ).
fof(t18_zfrefle1,axiom,
$true ).
fof(t19_zfrefle1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( r2_hidden(A,k2_relat_1(B))
=> v3_ordinal1(A) ) ) ).
fof(t20_zfrefle1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> r1_tarski(k2_relat_1(A),k8_ordinal2(A)) ) ).
fof(t21_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r4_zfrefle1(A,B)
& r4_zfrefle1(B,C) )
=> r4_zfrefle1(A,C) ) ) ) ) ).
fof(t22_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r4_zfrefle1(A,B)
=> r1_tarski(B,A) ) ) ) ).
fof(t23_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( ( r4_zfrefle1(A,B)
& r4_zfrefle1(B,A) )
=> A = B ) ) ) ).
fof(t24_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( ( v4_ordinal1(k1_relat_1(B))
& v2_ordinal2(B)
& r1_ordinal2(A,B) )
=> ( k1_relat_1(B) = k1_xboole_0
| r4_zfrefle1(A,k1_relat_1(B)) ) ) ) ) ).
fof(t25_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> r4_zfrefle1(k1_ordinal1(A),k4_ordinal2) ) ).
fof(t26_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( r4_zfrefle1(A,k1_ordinal1(B))
& ! [C] :
( v3_ordinal1(C)
=> A != k1_ordinal1(C) ) ) ) ) ).
fof(t27_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r4_zfrefle1(A,B)
=> ( v4_ordinal1(A)
<=> v4_ordinal1(B) ) ) ) ) ).
fof(t28_zfrefle1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( r4_zfrefle1(A,k1_xboole_0)
=> A = k1_xboole_0 ) ) ).
fof(t29_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ~ r4_zfrefle1(k2_ordinal2(A),B) ) ) ).
fof(t30_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( m2_ordinal4(C,A)
=> ~ ( r2_hidden(k5_ordinal2,A)
& v2_ordinal2(C)
& v3_ordinal2(C)
& ! [D] :
( m1_ordinal4(D,A)
=> ~ ( r2_hidden(B,D)
& k4_ordinal4(A,C,D) = D ) ) ) ) ) ) ).
fof(t31_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( m2_ordinal4(C,A)
=> ~ ( r2_hidden(k5_ordinal2,A)
& v2_ordinal2(C)
& v3_ordinal2(C)
& ! [D] :
( m1_ordinal4(D,A)
=> ~ ( r2_hidden(B,D)
& k4_ordinal4(A,C,D) = D
& r4_zfrefle1(D,k5_ordinal2) ) ) ) ) ) ) ).
fof(t32_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( m1_ordinal4(D,A)
=> ( r2_hidden(C,D)
=> r1_tarski(k5_zf_refle(A,B,C),k5_zf_refle(A,B,D)) ) ) )
& ! [C] :
( m1_ordinal4(C,A)
=> ( v4_ordinal1(C)
=> ( C = k1_xboole_0
| k5_zf_refle(A,B,C) = k3_card_3(k2_ordinal1(B,C)) ) ) )
& ! [C] :
( m2_ordinal4(C,A)
=> ~ ( v2_ordinal2(C)
& v3_ordinal2(C)
& ! [D] :
( m1_ordinal4(D,A)
=> ( k4_ordinal4(A,C,D) = D
=> ( k1_xboole_0 = D
| r3_zfrefle1(k5_zf_refle(A,B,D),k4_zf_refle(A,B)) ) ) ) ) ) ) ) ) ).
fof(t33_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> r2_hidden(k4_classes1(B),A) ) ) ).
fof(t34_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ( B != k1_xboole_0
=> ( ~ v1_xboole_0(k4_classes1(B))
& m1_subset_1(k4_classes1(B),A) ) ) ) ) ).
fof(t35_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [B] :
( m2_ordinal4(B,A)
=> ~ ( v2_ordinal2(B)
& v3_ordinal2(B)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ( ( k4_ordinal4(A,B,C) = C
& D = k4_classes1(C) )
=> ( k1_xboole_0 = C
| r3_zfrefle1(D,A) ) ) ) ) ) ) ) ) ).
fof(t36_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ~ ( r2_hidden(B,C)
& D = k4_classes1(C)
& r3_zfrefle1(D,A) ) ) ) ) ) ) ).
fof(t37_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r4_zfrefle1(B,k5_ordinal2)
& C = k4_classes1(B)
& r3_zfrefle1(C,A) ) ) ) ) ) ).
fof(t38_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( m1_ordinal4(D,A)
=> ( r2_hidden(C,D)
=> r1_tarski(k5_zf_refle(A,B,C),k5_zf_refle(A,B,D)) ) ) )
& ! [C] :
( m1_ordinal4(C,A)
=> ( v4_ordinal1(C)
=> ( C = k1_xboole_0
| k5_zf_refle(A,B,C) = k3_card_3(k2_ordinal1(B,C)) ) ) )
& ! [C] :
( m2_ordinal4(C,A)
=> ~ ( v2_ordinal2(C)
& v3_ordinal2(C)
& ! [D] :
( m1_ordinal4(D,A)
=> ( k4_ordinal4(A,C,D) = D
=> ( k1_xboole_0 = D
| r2_zfrefle1(k5_zf_refle(A,B,D),k4_zf_refle(A,B)) ) ) ) ) ) ) ) ) ).
fof(t39_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [B] :
( m2_ordinal4(B,A)
=> ~ ( v2_ordinal2(B)
& v3_ordinal2(B)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ( ( k4_ordinal4(A,B,C) = C
& D = k4_classes1(C) )
=> ( k1_xboole_0 = C
| r2_zfrefle1(D,A) ) ) ) ) ) ) ) ) ).
fof(t40_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ~ ( r2_hidden(B,C)
& D = k4_classes1(C)
& r2_zfrefle1(D,A) ) ) ) ) ) ) ).
fof(t41_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r4_zfrefle1(B,k5_ordinal2)
& C = k4_classes1(B)
& r2_zfrefle1(C,A) ) ) ) ) ) ).
fof(t42_zfrefle1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ ( r2_hidden(k5_ordinal2,A)
& ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r4_zfrefle1(B,k5_ordinal2)
& C = k4_classes1(B)
& v1_zf_model(C) ) ) ) ) ) ).
fof(t43_zfrefle1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ~ ( r2_hidden(k5_ordinal2,B)
& r2_hidden(A,B)
& ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r2_hidden(A,C)
& r2_hidden(C,B)
& v1_zf_model(C) ) ) ) ) ).
fof(s1_zfrefle1,axiom,
( ! [A] :
~ ( r2_hidden(A,f1_s1_zfrefle1)
& ! [B] : ~ p1_s1_zfrefle1(A,B) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_zfrefle1
& ! [B] :
( r2_hidden(B,f1_s1_zfrefle1)
=> p1_s1_zfrefle1(B,k1_funct_1(A,B)) ) ) ) ).
fof(symmetry_r2_zfrefle1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( r2_zfrefle1(A,B)
=> r2_zfrefle1(B,A) ) ) ).
fof(reflexivity_r2_zfrefle1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> r2_zfrefle1(A,A) ) ).
fof(reflexivity_r3_zfrefle1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> r3_zfrefle1(A,A) ) ).
fof(reflexivity_r4_zfrefle1,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> r4_zfrefle1(A,A) ) ).
fof(dt_k1_zfrefle1,axiom,
$true ).
fof(dt_k2_zfrefle1,axiom,
m1_subset_1(k2_zfrefle1,k1_zfmisc_1(k9_zf_lang)) ).
fof(redefinition_k2_zfrefle1,axiom,
k2_zfrefle1 = k1_zfrefle1 ).
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