SET007 Axioms: SET007+863.ax
%------------------------------------------------------------------------------
% File : SET007+863 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Consequences of the Sequent Calculus
% 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 : calcul_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 1 unt; 0 def)
% Number of atoms : 217 ( 30 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 178 ( 8 ~; 2 |; 48 &)
% ( 6 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 42 ( 42 usr; 8 con; 0-6 aty)
% Number of variables : 112 ( 110 !; 2 ?)
% 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(fc1_calcul_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> v1_finset_1(k1_calcul_2(A,B)) ) ).
fof(fc2_calcul_2,axiom,
! [A] :
( m1_finseq_1(A,k7_cqc_lang)
=> ( v1_finset_1(k1_card_1(A))
& v1_card_1(k1_card_1(A)) ) ) ).
fof(t1_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(A,k2_calcul_2(B,C))
<=> ( r1_xreal_0(k2_xcmplx_0(np__1,B),A)
& r1_xreal_0(A,k2_xcmplx_0(C,B)) ) ) ) ) ) ).
fof(t2_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> k2_calcul_2(A,np__0) = k1_xboole_0 ) ).
fof(t3_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A = np__0
| r2_hidden(k2_xcmplx_0(A,B),k2_calcul_2(B,A)) ) ) ) ).
fof(t4_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r1_xreal_0(A,B)
<=> r1_tarski(k2_calcul_2(C,A),k2_calcul_2(C,B)) ) ) ) ) ).
fof(t5_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k2_xboole_0(k2_calcul_2(A,B),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(A,B),np__1))) = k2_calcul_2(A,k2_xcmplx_0(B,np__1)) ) ) ).
fof(t6_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_wellord2(k2_calcul_2(A,B),B) ) ) ).
fof(t7_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_tarski(k2_calcul_2(A,B),k2_finseq_1(k1_nat_1(A,B))) ) ) ).
fof(t8_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xboole_0(k2_finseq_1(A),k2_calcul_2(A,B)) ) ) ).
fof(t9_calcul_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k2_finseq_1(k3_finseq_1(k7_finseq_1(A,B))) = k4_subset_1(k5_numbers,k2_finseq_1(k3_finseq_1(A)),k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B))) ) ) ).
fof(t10_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k3_finseq_1(k14_finseq_1(k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)))) = k3_finseq_1(B) ) ) ).
fof(t11_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k4_relset_1(k5_numbers,k5_numbers,k14_finseq_1(k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)))) = k4_relset_1(k5_numbers,k7_cqc_lang,B) ) ) ).
fof(t12_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k5_relset_1(k5_numbers,k5_numbers,k14_finseq_1(k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)))) = k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)) ) ) ).
fof(t13_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r2_hidden(A,k4_relset_1(k5_numbers,k5_numbers,k14_finseq_1(k2_calcul_2(k3_finseq_1(B),k3_finseq_1(C)))))
=> k1_funct_1(k14_finseq_1(k2_calcul_2(k3_finseq_1(B),k3_finseq_1(C))),A) = k1_nat_1(k3_finseq_1(B),A) ) ) ) ) ).
fof(t14_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> r1_tarski(k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)),k4_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B))) ) ) ).
fof(t15_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k4_relset_1(k5_numbers,k7_cqc_lang,k8_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B),k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)))) = k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)) ) ) ).
fof(t16_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k15_finseq_1(k8_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B),k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B)))) = k7_relset_1(k5_numbers,k5_numbers,k5_numbers,k7_cqc_lang,k14_finseq_1(k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B))),k8_finseq_1(k7_cqc_lang,A,B)) ) ) ).
fof(t17_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> k4_finseq_1(k15_finseq_1(k8_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B),k2_calcul_2(k3_finseq_1(A),k3_finseq_1(B))))) = k4_relset_1(k5_numbers,k7_cqc_lang,B) ) ) ).
fof(t18_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> r2_calcul_1(A,k8_finseq_1(k7_cqc_lang,B,A)) ) ) ).
fof(d2_calcul_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& v3_funct_2(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& m2_relset_1(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B)) )
=> k3_calcul_2(A,B,C) = k7_relset_1(k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B),k5_numbers,A,C,B) ) ) ) ).
fof(t19_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k4_relset_1(k5_numbers,k7_cqc_lang,A),k4_relset_1(k5_numbers,k7_cqc_lang,A))
& v3_funct_2(B,k4_relset_1(k5_numbers,k7_cqc_lang,A),k4_relset_1(k5_numbers,k7_cqc_lang,A))
& m2_relset_1(B,k4_relset_1(k5_numbers,k7_cqc_lang,A),k4_relset_1(k5_numbers,k7_cqc_lang,A)) )
=> k4_relset_1(k5_numbers,k7_cqc_lang,k3_calcul_2(k7_cqc_lang,A,B)) = k4_relset_1(k5_numbers,k7_cqc_lang,A) ) ) ).
fof(t20_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,B),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ).
fof(d3_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ( r1_xreal_0(np__1,k3_finseq_1(A))
=> ( B = k4_calcul_2(A)
<=> B = k1_funct_1(A,np__1) ) )
& ( ~ r1_xreal_0(np__1,k3_finseq_1(A))
=> ( B = k4_calcul_2(A)
<=> B = k9_cqc_lang ) ) ) ) ) ).
fof(d4_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( B = k5_calcul_2(A)
<=> ? [C] :
( m2_finseq_1(C,k7_cqc_lang)
& B = k1_funct_1(C,k3_finseq_1(A))
& k3_finseq_1(C) = k3_finseq_1(A)
& ( k1_funct_1(C,np__1) = k4_calcul_2(A)
| k3_finseq_1(A) = np__0 )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& ~ r1_xreal_0(k3_finseq_1(A),D)
& ! [E] :
( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang)
=> ! [F] :
( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang)
=> ~ ( E = k1_funct_1(A,k1_nat_1(D,np__1))
& F = k1_funct_1(C,D)
& k1_funct_1(C,k1_nat_1(D,np__1)) = k12_cqc_lang(E,F) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,A))) ) ) ).
fof(t22_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k11_cqc_lang(A,B))))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ).
fof(t23_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k11_cqc_lang(A,B))))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ).
fof(t24_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)))
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B))) )
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ).
fof(t25_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)))
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A)))) )
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ).
fof(t26_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B)))
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A))),k12_finseq_1(k7_cqc_lang,B))) )
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ).
fof(t27_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B)))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k12_cqc_lang(A,B)))) ) ) ) ) ).
fof(t28_calcul_2,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ( ( r1_xreal_0(np__1,k3_finseq_1(A))
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,A)) )
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,k5_calcul_2(k4_finseq_5(k7_cqc_lang,A))))) ) ) ) ).
fof(t29_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B))
& v3_funct_2(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B))
& m2_relset_1(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B)) )
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k3_calcul_2(k7_cqc_lang,B,C),k12_finseq_1(k7_cqc_lang,k5_calcul_2(k4_finseq_5(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)))))))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k3_calcul_2(k7_cqc_lang,B,C),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ).
fof(t30_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B))
& v3_funct_2(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B))
& m2_relset_1(C,k4_relset_1(k5_numbers,k7_cqc_lang,B),k4_relset_1(k5_numbers,k7_cqc_lang,B)) )
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k3_calcul_2(k7_cqc_lang,B,C),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ).
fof(t31_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( r1_xreal_0(np__1,A)
=> k2_relat_1(k2_finseq_2(A,B)) = k2_relat_1(k9_finseq_1(B)) ) ) ).
fof(t32_calcul_2,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k7_cqc_lang)
=> ( ( r1_xreal_0(np__1,C)
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k6_calcul_2(k7_cqc_lang,C,A)),k12_finseq_1(k7_cqc_lang,B))) )
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ) ).
fof(dt_k1_calcul_2,axiom,
$true ).
fof(dt_k2_calcul_2,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m1_subset_1(k2_calcul_2(A,B),k1_zfmisc_1(k5_numbers)) ) ).
fof(redefinition_k2_calcul_2,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> k2_calcul_2(A,B) = k1_calcul_2(A,B) ) ).
fof(dt_k3_calcul_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& v3_funct_2(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& m1_relset_1(C,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B)) )
=> m2_finseq_1(k3_calcul_2(A,B,C),A) ) ).
fof(dt_k4_calcul_2,axiom,
! [A] :
( m1_finseq_1(A,k7_cqc_lang)
=> m2_subset_1(k4_calcul_2(A),k8_qc_lang1,k7_cqc_lang) ) ).
fof(dt_k5_calcul_2,axiom,
! [A] :
( m1_finseq_1(A,k7_cqc_lang)
=> m2_subset_1(k5_calcul_2(A),k8_qc_lang1,k7_cqc_lang) ) ).
fof(dt_k6_calcul_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> m2_finseq_1(k6_calcul_2(A,B,C),A) ) ).
fof(redefinition_k6_calcul_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> k6_calcul_2(A,B,C) = k2_finseq_2(B,C) ) ).
fof(d1_calcul_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k1_calcul_2(A,B) = a_2_0_calcul_2(A,B) ) ) ).
fof(fraenkel_a_2_0_calcul_2,axiom,
! [A,B,C] :
( ( v4_ordinal2(B)
& v4_ordinal2(C) )
=> ( r2_hidden(A,a_2_0_calcul_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = D
& r1_xreal_0(k2_xcmplx_0(np__1,B),D)
& r1_xreal_0(D,k2_xcmplx_0(C,B)) ) ) ) ).
%------------------------------------------------------------------------------