SET007 Axioms: SET007+828.ax
%------------------------------------------------------------------------------
% File : SET007+828 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Complex Valued Function's Space
% 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 : cfuncdom [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 1 unt; 0 def)
% Number of atoms : 403 ( 118 equ)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 421 ( 79 ~; 2 |; 168 &)
% ( 7 <=>; 165 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 28 ( 27 usr; 0 prp; 1-4 aty)
% Number of functors : 42 ( 42 usr; 6 con; 0-6 aty)
% Number of variables : 193 ( 173 !; 20 ?)
% 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_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_cfuncdom(A))
& v3_vectsp_1(k7_cfuncdom(A)) ) ) ).
fof(fc2_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_cfuncdom(A))
& v2_group_1(k7_cfuncdom(A))
& v3_vectsp_1(k7_cfuncdom(A))
& v6_vectsp_1(k7_cfuncdom(A))
& v8_vectsp_1(k7_cfuncdom(A)) ) ) ).
fof(rc1_cfuncdom,axiom,
? [A] :
( l1_cfuncdom(A)
& v1_cfuncdom(A) ) ).
fof(rc2_cfuncdom,axiom,
? [A] :
( l1_cfuncdom(A)
& ~ v3_struct_0(A) ) ).
fof(fc3_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k8_cfuncdom(A))
& v1_cfuncdom(k8_cfuncdom(A)) ) ) ).
fof(fc4_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k8_cfuncdom(A))
& v2_group_1(k8_cfuncdom(A))
& v6_vectsp_1(k8_cfuncdom(A))
& v8_vectsp_1(k8_cfuncdom(A))
& v1_cfuncdom(k8_cfuncdom(A)) ) ) ).
fof(rc3_cfuncdom,axiom,
? [A] :
( l1_cfuncdom(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v1_cfuncdom(A)
& v2_cfuncdom(A) ) ).
fof(d1_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
=> ( B = k1_cfuncdom(A)
<=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,B,C,D) = k4_funcsdom(k2_numbers,A,k27_binop_2,C,D) ) ) ) ) ) ).
fof(d2_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
=> ( B = k2_cfuncdom(A)
<=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,B,C,D) = k4_funcsdom(k2_numbers,A,k29_binop_2,C,D) ) ) ) ) ) ).
fof(d3_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(B,k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
=> ( B = k3_cfuncdom(A)
<=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k2_numbers,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),B,k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,D)),E) = k9_complex1(C,k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) ).
fof(d4_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_cfuncdom(A) = k2_funcop_1(A,np__0) ) ).
fof(d5_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k5_cfuncdom(A) = k2_funcop_1(A,k6_complex1) ) ).
fof(t1_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ( B = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k2_numbers,B,E) = k8_complex1(k8_funct_2(A,k2_numbers,C,E),k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) ).
fof(t2_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ( B = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,D)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k2_numbers,B,E) = k9_complex1(k8_funct_2(A,k2_numbers,C,E),k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) ).
fof(t3_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k8_funct_2(A,k2_numbers,k5_cfuncdom(A),B) = k6_complex1 ) ) ).
fof(t4_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k8_funct_2(A,k2_numbers,k4_cfuncdom(A),B) = np__0 ) ) ).
fof(t5_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_cfuncdom(A) != k5_cfuncdom(A) ) ).
fof(t6_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> ( B = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,C))
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k2_numbers,B,E) = k9_complex1(D,k8_funct_2(A,k2_numbers,C,E)) ) ) ) ) ) ) ).
fof(t7_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,C) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,B) ) ) ) ).
fof(t8_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,C),D) ) ) ) ) ).
fof(t9_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C) = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,B) ) ) ) ).
fof(t10_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C),D) ) ) ) ) ).
fof(t11_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k5_cfuncdom(A),B) = B ) ) ).
fof(t12_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k4_cfuncdom(A),B) = B ) ) ).
fof(t13_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k10_complex1(k6_complex1),B))) = k4_cfuncdom(A) ) ) ).
fof(t14_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k6_complex1,B)) = B ) ) ).
fof(t15_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B)))) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k9_complex1(C,D),B)) ) ) ) ) ).
fof(t16_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,B)),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B))) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k8_complex1(C,D),B)) ) ) ) ) ).
fof(t17_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,D)) ) ) ) ) ).
fof(t18_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [C] :
( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B)),C) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C))) ) ) ) ) ).
fof(t19_cfuncdom,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ? [C] :
( m2_fraenkel(C,B,k2_numbers,k1_fraenkel(B,k2_numbers))
& ? [D] :
( m2_fraenkel(D,B,k2_numbers,k1_fraenkel(B,k2_numbers))
& ! [E] :
( r2_hidden(E,B)
=> ( ( E = A
=> k1_funct_1(C,E) = k6_complex1 )
& ( E != A
=> k1_funct_1(C,E) = np__0 ) ) )
& ! [E] :
( r2_hidden(E,B)
=> ( ( E = A
=> k1_funct_1(D,E) = np__0 )
& ( E != A
=> k1_funct_1(D,E) = k6_complex1 ) ) ) ) ) ) ).
fof(t20_cfuncdom,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ! [E] :
( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ( ( r2_hidden(A,C)
& r2_hidden(B,C)
& ! [F] :
( r2_hidden(F,C)
=> ( ( F = A
=> k1_funct_1(D,F) = k6_complex1 )
& ( F != A
=> k1_funct_1(D,F) = np__0 ) ) )
& ! [F] :
( r2_hidden(F,C)
=> ( ( F = A
=> k1_funct_1(E,F) = np__0 )
& ( F != A
=> k1_funct_1(E,F) = k6_complex1 ) ) ) )
=> ( A = B
| ! [F] :
( m1_subset_1(F,k2_numbers)
=> ! [G] :
( m1_subset_1(G,k2_numbers)
=> ( k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
=> ( F = k5_complex1
& G = k5_complex1 ) ) ) ) ) ) ) ) ) ).
fof(t21_cfuncdom,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ~ ( r2_hidden(A,C)
& r2_hidden(B,C)
& A != B
& ! [D] :
( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ! [E] :
( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ? [F] :
( m1_subset_1(F,k2_numbers)
& ? [G] :
( m1_subset_1(G,k2_numbers)
& k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
& ~ ( F = np__0
& G = np__0 ) ) ) ) ) ) ) ).
fof(t22_cfuncdom,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ! [E] :
( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ( ( C = k2_tarski(A,B)
& ! [F] :
( r2_hidden(F,C)
=> ( ( F = A
=> k1_funct_1(D,F) = k6_complex1 )
& ( F != A
=> k1_funct_1(D,F) = np__0 ) ) )
& ! [F] :
( r2_hidden(F,C)
=> ( ( F = A
=> k1_funct_1(E,F) = np__0 )
& ( F != A
=> k1_funct_1(E,F) = k6_complex1 ) ) ) )
=> ( A = B
| ! [F] :
( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ? [G] :
( m1_subset_1(G,k2_numbers)
& ? [H] :
( m1_subset_1(H,k2_numbers)
& F = k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) ) ).
fof(t23_cfuncdom,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ~ ( C = k2_tarski(A,B)
& A != B
& ! [D] :
( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ! [E] :
( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ? [F] :
( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
& ! [G] :
( m1_subset_1(G,k2_numbers)
=> ! [H] :
( m1_subset_1(H,k2_numbers)
=> F != k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) ).
fof(t24_cfuncdom,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ~ ( C = k2_tarski(A,B)
& A != B
& ! [D] :
( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ! [E] :
( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ~ ( ! [F] :
( m1_subset_1(F,k2_numbers)
=> ! [G] :
( m1_subset_1(G,k2_numbers)
=> ( k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
=> ( F = np__0
& G = np__0 ) ) ) )
& ! [F] :
( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
=> ? [G] :
( m1_subset_1(G,k2_numbers)
& ? [H] :
( m1_subset_1(H,k2_numbers)
& F = k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) ) ).
fof(t25_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& v3_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& v4_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& v5_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& v6_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& v2_clvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
& l1_clvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A))) ) ) ).
fof(d6_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k6_cfuncdom(A) = g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)) ) ).
fof(t26_cfuncdom,axiom,
? [A] :
( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v1_clvect_1(A)
& v2_clvect_1(A)
& l1_clvect_1(A)
& ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ! [D] :
( m1_subset_1(D,k2_numbers)
=> ! [E] :
( m1_subset_1(E,k2_numbers)
=> ( k4_rlvect_1(A,k1_clvect_1(A,B,D),k1_clvect_1(A,C,E)) = k1_rlvect_1(A)
=> ( D = np__0
& E = np__0 ) ) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ? [E] :
( m1_subset_1(E,k2_numbers)
& ? [F] :
( m1_subset_1(F,k2_numbers)
& D = k4_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,C,F)) ) ) ) ) ) ) ).
fof(d7_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k7_cfuncdom(A) = g3_vectsp_1(k1_fraenkel(A,k2_numbers),k1_cfuncdom(A),k2_cfuncdom(A),k5_cfuncdom(A),k4_cfuncdom(A)) ) ).
fof(t27_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k7_cfuncdom(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k7_cfuncdom(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k7_cfuncdom(A)))
=> ( k2_rlvect_1(k7_cfuncdom(A),B,C) = k2_rlvect_1(k7_cfuncdom(A),C,B)
& k2_rlvect_1(k7_cfuncdom(A),k2_rlvect_1(k7_cfuncdom(A),B,C),D) = k2_rlvect_1(k7_cfuncdom(A),B,k2_rlvect_1(k7_cfuncdom(A),C,D))
& k2_rlvect_1(k7_cfuncdom(A),B,k1_rlvect_1(k7_cfuncdom(A))) = B
& ? [E] :
( m1_subset_1(E,u1_struct_0(k7_cfuncdom(A)))
& k2_rlvect_1(k7_cfuncdom(A),B,E) = k1_rlvect_1(k7_cfuncdom(A)) )
& k1_group_1(k7_cfuncdom(A),B,C) = k1_group_1(k7_cfuncdom(A),C,B)
& k1_group_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),B,C),D) = k1_group_1(k7_cfuncdom(A),B,k1_group_1(k7_cfuncdom(A),C,D))
& k1_group_1(k7_cfuncdom(A),B,k2_group_1(k7_cfuncdom(A))) = B
& k1_group_1(k7_cfuncdom(A),k2_group_1(k7_cfuncdom(A)),B) = B
& k1_group_1(k7_cfuncdom(A),B,k2_rlvect_1(k7_cfuncdom(A),C,D)) = k2_rlvect_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),B,C),k1_group_1(k7_cfuncdom(A),B,D))
& k1_group_1(k7_cfuncdom(A),k2_rlvect_1(k7_cfuncdom(A),C,D),B) = k2_rlvect_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),C,B),k1_group_1(k7_cfuncdom(A),D,B)) ) ) ) ) ) ).
fof(t28_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_cfuncdom(A))
& v3_rlvect_1(k7_cfuncdom(A))
& v4_rlvect_1(k7_cfuncdom(A))
& v5_rlvect_1(k7_cfuncdom(A))
& v6_rlvect_1(k7_cfuncdom(A))
& v4_group_1(k7_cfuncdom(A))
& v7_group_1(k7_cfuncdom(A))
& v6_vectsp_1(k7_cfuncdom(A))
& v7_vectsp_1(k7_cfuncdom(A))
& v8_vectsp_1(k7_cfuncdom(A))
& l3_vectsp_1(k7_cfuncdom(A)) ) ) ).
fof(d8_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k8_cfuncdom(A) = g1_cfuncdom(k1_fraenkel(A,k2_numbers),k2_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A),k5_cfuncdom(A),k4_cfuncdom(A)) ) ).
fof(t29_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k8_cfuncdom(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k8_cfuncdom(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k8_cfuncdom(A)))
=> ! [E] :
( m1_subset_1(E,k2_numbers)
=> ! [F] :
( m1_subset_1(F,k2_numbers)
=> ( k2_rlvect_1(k8_cfuncdom(A),B,C) = k2_rlvect_1(k8_cfuncdom(A),C,B)
& k2_rlvect_1(k8_cfuncdom(A),k2_rlvect_1(k8_cfuncdom(A),B,C),D) = k2_rlvect_1(k8_cfuncdom(A),B,k2_rlvect_1(k8_cfuncdom(A),C,D))
& k2_rlvect_1(k8_cfuncdom(A),B,k1_rlvect_1(k8_cfuncdom(A))) = B
& ? [G] :
( m1_subset_1(G,u1_struct_0(k8_cfuncdom(A)))
& k2_rlvect_1(k8_cfuncdom(A),B,G) = k1_rlvect_1(k8_cfuncdom(A)) )
& k1_group_1(k8_cfuncdom(A),B,C) = k1_group_1(k8_cfuncdom(A),C,B)
& k1_group_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),D) = k1_group_1(k8_cfuncdom(A),B,k1_group_1(k8_cfuncdom(A),C,D))
& k1_group_1(k8_cfuncdom(A),B,k2_group_1(k8_cfuncdom(A))) = B
& k1_group_1(k8_cfuncdom(A),B,k2_rlvect_1(k8_cfuncdom(A),C,D)) = k2_rlvect_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),k1_group_1(k8_cfuncdom(A),B,D))
& k1_clvect_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),E) = k1_group_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),C)
& k1_clvect_1(k8_cfuncdom(A),k2_rlvect_1(k8_cfuncdom(A),B,C),E) = k2_rlvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),k1_clvect_1(k8_cfuncdom(A),C,E))
& k1_clvect_1(k8_cfuncdom(A),B,k8_complex1(E,F)) = k2_rlvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),k1_clvect_1(k8_cfuncdom(A),B,F))
& k1_clvect_1(k8_cfuncdom(A),B,k9_complex1(E,F)) = k1_clvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,F),E) ) ) ) ) ) ) ) ).
fof(d9_cfuncdom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_cfuncdom(A) )
=> ( v2_cfuncdom(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k2_numbers)
=> ! [F] :
( m1_subset_1(F,k2_numbers)
=> ( k1_group_1(A,B,k2_group_1(A)) = B
& k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D))
& k1_clvect_1(A,k1_group_1(A,B,C),E) = k1_group_1(A,k1_clvect_1(A,B,E),C)
& k1_clvect_1(A,k2_rlvect_1(A,B,C),E) = k2_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,C,E))
& k1_clvect_1(A,B,k8_complex1(E,F)) = k2_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,B,F))
& k1_clvect_1(A,B,k9_complex1(E,F)) = k1_clvect_1(A,k1_clvect_1(A,B,F),E) ) ) ) ) ) ) ) ) ).
fof(t30_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k8_cfuncdom(A))
& v3_rlvect_1(k8_cfuncdom(A))
& v4_rlvect_1(k8_cfuncdom(A))
& v5_rlvect_1(k8_cfuncdom(A))
& v6_rlvect_1(k8_cfuncdom(A))
& v4_group_1(k8_cfuncdom(A))
& v7_group_1(k8_cfuncdom(A))
& v2_cfuncdom(k8_cfuncdom(A))
& l1_cfuncdom(k8_cfuncdom(A)) ) ) ).
fof(t31_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k2_group_1(k7_cfuncdom(A)) = k5_cfuncdom(A) ) ).
fof(t32_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k2_group_1(k8_cfuncdom(A)) = k5_cfuncdom(A) ) ).
fof(dt_l1_cfuncdom,axiom,
! [A] :
( l1_cfuncdom(A)
=> ( l3_vectsp_1(A)
& l1_clvect_1(A) ) ) ).
fof(existence_l1_cfuncdom,axiom,
? [A] : l1_cfuncdom(A) ).
fof(abstractness_v1_cfuncdom,axiom,
! [A] :
( l1_cfuncdom(A)
=> ( v1_cfuncdom(A)
=> A = g1_cfuncdom(u1_struct_0(A),u1_group_1(A),u1_rlvect_1(A),u1_clvect_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ).
fof(dt_k1_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k1_cfuncdom(A))
& v1_funct_2(k1_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(k1_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) ).
fof(dt_k2_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k2_cfuncdom(A))
& v1_funct_2(k2_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(k2_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) ).
fof(dt_k3_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k3_cfuncdom(A))
& v1_funct_2(k3_cfuncdom(A),k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
& m2_relset_1(k3_cfuncdom(A),k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) ).
fof(dt_k4_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m2_fraenkel(k4_cfuncdom(A),A,k2_numbers,k1_fraenkel(A,k2_numbers)) ) ).
fof(dt_k5_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m2_fraenkel(k5_cfuncdom(A),A,k2_numbers,k1_fraenkel(A,k2_numbers)) ) ).
fof(dt_k6_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k6_cfuncdom(A))
& v3_rlvect_1(k6_cfuncdom(A))
& v4_rlvect_1(k6_cfuncdom(A))
& v5_rlvect_1(k6_cfuncdom(A))
& v6_rlvect_1(k6_cfuncdom(A))
& v1_clvect_1(k6_cfuncdom(A))
& v2_clvect_1(k6_cfuncdom(A))
& l1_clvect_1(k6_cfuncdom(A)) ) ) ).
fof(dt_k7_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v3_vectsp_1(k7_cfuncdom(A))
& l3_vectsp_1(k7_cfuncdom(A)) ) ) ).
fof(dt_k8_cfuncdom,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_cfuncdom(k8_cfuncdom(A))
& l1_cfuncdom(k8_cfuncdom(A)) ) ) ).
fof(dt_g1_cfuncdom,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(k2_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ( v1_cfuncdom(g1_cfuncdom(A,B,C,D,E,F))
& l1_cfuncdom(g1_cfuncdom(A,B,C,D,E,F)) ) ) ).
fof(free_g1_cfuncdom,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(k2_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ! [G,H,I,J,K,L] :
( g1_cfuncdom(A,B,C,D,E,F) = g1_cfuncdom(G,H,I,J,K,L)
=> ( A = G
& B = H
& C = I
& D = J
& E = K
& F = L ) ) ) ).
%------------------------------------------------------------------------------