SET007 Axioms: SET007+374.ax
%------------------------------------------------------------------------------
% File : SET007+374 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hahn-Banach 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 : hahnban [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 6 unt; 0 def)
% Number of atoms : 668 ( 60 equ)
% Maximal formula atoms : 30 ( 11 avg)
% Number of connectives : 680 ( 68 ~; 6 |; 366 &)
% ( 9 <=>; 231 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 11 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 34 ( 32 usr; 1 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 6 con; 0-4 aty)
% Number of variables : 204 ( 195 !; 9 ?)
% 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(rc1_hahnban,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v6_ordinal1(B) ) ) ).
fof(cc1_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v2_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v1_hahnban(B,A) ) ) ) ) ).
fof(cc2_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v3_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v4_hahnban(B,A) ) ) ) ) ).
fof(cc3_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v5_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v4_hahnban(B,A) ) ) ) ) ).
fof(cc4_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v5_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v7_hahnban(B,A) ) ) ) ) ).
fof(cc5_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v6_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v5_hahnban(B,A) ) ) ) ) ).
fof(cc6_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v4_hahnban(B,A)
& v7_hahnban(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v5_hahnban(B,A) ) ) ) ) ).
fof(rc2_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),k6_supinf_1)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v1_hahnban(B,A)
& v2_hahnban(B,A)
& v3_hahnban(B,A)
& v4_hahnban(B,A)
& v5_hahnban(B,A)
& v6_hahnban(B,A)
& v7_hahnban(B,A) ) ) ).
fof(t1_hahnban,axiom,
$true ).
fof(t2_hahnban,axiom,
$true ).
fof(t3_hahnban,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
~ ( A != k1_tarski(B)
& ! [C] :
( m1_subset_1(C,A)
=> C = B ) ) ) ).
fof(t4_hahnban,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k4_rfunct_3(A,B))) )
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ).
fof(t6_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
=> r1_supinf_1(B,k5_supinf_1) ) )
=> A = k11_supinf_1(k5_supinf_1) ) ) ).
fof(t7_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
=> r1_supinf_1(k4_supinf_1,B) ) )
=> A = k11_supinf_1(k4_supinf_1) ) ) ).
fof(t8_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(k9_supinf_1(A),B)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,A)
& ~ r1_supinf_1(C,B) ) ) ) ) ) ).
fof(t9_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,k10_supinf_1(A))
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,A)
& ~ r1_supinf_1(B,C) ) ) ) ) ) ).
fof(t10_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k3_supinf_1)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,B) )
=> r1_supinf_1(C,D) ) ) )
=> r1_supinf_1(k9_supinf_1(A),k10_supinf_1(B)) ) ) ) ).
fof(t11_hahnban,axiom,
$true ).
fof(t12_hahnban,axiom,
$true ).
fof(t13_hahnban,axiom,
! [A,B,C] :
( ( v6_ordinal1(C)
& m1_subset_1(C,k1_zfmisc_1(k4_rfunct_3(A,B))) )
=> r2_hidden(k3_tarski(C),k4_rfunct_3(A,B)) ) ).
fof(t14_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> r1_tarski(u1_struct_0(B),u1_struct_0(k1_rlsub_2(A,B,C))) ) ) ) ).
fof(t15_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r1_rlvect_1(B,E)
& r1_rlvect_1(C,F)
& D = k4_rlvect_1(A,E,F) )
=> k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F) ) ) ) ) ) ) ) ) ).
fof(t16_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F)
=> D = k4_rlvect_1(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t17_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F)
=> ( r1_rlvect_1(B,E)
& r1_rlvect_1(C,F) ) ) ) ) ) ) ) ) ) ).
fof(t18_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F)
=> k4_rlsub_2(A,D,C,B) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),F,E) ) ) ) ) ) ) ) ) ).
fof(t19_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_rlvect_1(B,D)
=> k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),D,k1_rlvect_1(A)) ) ) ) ) ) ) ).
fof(t20_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( r1_rlsub_2(A,B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_rlvect_1(C,D)
=> k4_rlsub_2(A,D,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),k1_rlvect_1(A),D) ) ) ) ) ) ) ).
fof(t21_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_rlvect_1(C,D)
=> m1_subset_1(D,u1_struct_0(B)) ) ) ) ) ) ).
fof(t22_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_rlsub_1(D,A)
=> ! [E] :
( m1_rlsub_1(E,D)
=> ! [F] :
( m1_rlsub_1(F,D)
=> ( ( E = B
& F = C )
=> k1_rlsub_2(D,E,F) = k1_rlsub_2(A,B,C) ) ) ) ) ) ) ) ).
fof(t23_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( C = D
=> k1_rlvect_3(B,k6_domain_1(u1_struct_0(B),D)) = k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),C)) ) ) ) ) ) ).
fof(t24_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ( ~ r1_rlvect_1(C,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& E = C )
=> r1_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) ) ) ) ) ) ) ) ).
fof(t25_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& C = E )
=> ( r1_rlvect_1(C,B)
| k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),D,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(E),u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),k1_rlvect_1(E),D) ) ) ) ) ) ) ) ).
fof(t26_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& C = E )
=> ( r1_rlvect_1(C,B)
| ! [F] :
( m1_subset_1(F,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ( r1_rlvect_1(C,F)
=> k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),u1_struct_0(A),F,k1_rlvect_1(A)) ) ) ) ) ) ) ) ) ) ).
fof(t27_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_rlsub_1(D,A)
=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& k4_rlsub_2(A,B,C,D) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F) ) ) ) ) ) ) ).
fof(t28_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& C = E )
=> ( r1_rlvect_1(C,B)
| ! [F] :
( m1_subset_1(F,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ? [G] :
( m1_subset_1(G,u1_struct_0(C))
& ? [H] :
( m1_subset_1(H,k1_numbers)
& k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),G,k3_rlvect_1(A,B,H)) ) ) ) ) ) ) ) ) ) ) ).
fof(t29_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& C = E )
=> ( r1_rlvect_1(C,B)
| ! [F] :
( m1_subset_1(F,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(C))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(C))
=> ! [J] :
( m1_subset_1(J,k1_numbers)
=> ! [K] :
( m1_subset_1(K,k1_numbers)
=> ( ( k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),H,k3_rlvect_1(A,B,J))
& k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),G,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),I,k3_rlvect_1(A,B,K)) )
=> k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k4_rlvect_1(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,G),E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),k4_rlvect_1(C,H,I),k3_rlvect_1(A,B,k3_real_1(J,K))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlsub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [E] :
( m1_rlsub_1(E,k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))))
=> ( ( B = D
& C = E )
=> ( r1_rlvect_1(C,B)
| ! [F] :
( m1_subset_1(F,u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ! [I] :
( m1_subset_1(I,k1_numbers)
=> ( k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),G,k3_rlvect_1(A,B,I))
=> k4_rlsub_2(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k3_rlvect_1(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),F,H),E,k1_rlvect_3(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B))),k6_domain_1(u1_struct_0(k1_rlsub_2(A,C,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),B)))),D))) = k1_domain_1(u1_struct_0(C),u1_struct_0(A),k3_rlvect_1(C,G,H),k3_rlvect_1(A,B,k4_real_1(H,I))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d1_hahnban,axiom,
$true ).
fof(d2_hahnban,axiom,
$true ).
fof(d3_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v1_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k4_rlvect_1(A,C,D)),k3_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D))) ) ) ) ) ) ).
fof(d4_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v2_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k4_rlvect_1(A,C,D)) = k3_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)) ) ) ) ) ) ).
fof(d5_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v3_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k3_rlvect_1(A,C,D)) = k4_real_1(D,k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C)) ) ) ) ) ) ).
fof(d6_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v4_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k3_rlvect_1(A,C,D)) = k4_real_1(D,k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C)) ) ) ) ) ) ) ).
fof(d7_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v5_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r1_xreal_0(np__0,D)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k3_rlvect_1(A,C,D)) = k4_real_1(D,k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C)) ) ) ) ) ) ) ).
fof(d8_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v6_hahnban(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k3_rlvect_1(A,C,D)) = k4_real_1(k18_complex1(D),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C)) ) ) ) ) ) ).
fof(d9_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ( v7_hahnban(B,A)
<=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k1_rlvect_1(A)) = np__0 ) ) ) ).
fof(t31_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v3_hahnban(B,A)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k5_rlvect_1(A,C)) = k1_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C)) ) ) ) ).
fof(t32_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v2_hahnban(B,A)
& v3_hahnban(B,A)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k6_rlvect_1(A,C,D)) = k5_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)) ) ) ) ) ).
fof(t33_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
& v2_hahnban(B,A)
& m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k1_rlvect_1(A)) = np__0 ) ) ).
fof(t34_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),k6_supinf_1)
& v2_hahnban(C,B)
& v3_hahnban(C,B)
& m2_relset_1(C,u1_struct_0(B),k6_supinf_1) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D)))))
=> ( D = E
=> ( r1_rlvect_1(B,D)
| ! [F] :
( m1_subset_1(F,k1_numbers)
=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D)))),k6_supinf_1)
& v2_hahnban(G,k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D))))
& v3_hahnban(G,k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D))))
& m2_relset_1(G,u1_struct_0(k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D)))),k6_supinf_1)
& k2_partfun1(u1_struct_0(k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D)))),k6_supinf_1,G,u1_struct_0(B)) = C
& k8_funct_2(u1_struct_0(k1_rlsub_2(A,B,k1_rlvect_3(A,k6_domain_1(u1_struct_0(A),D)))),k6_supinf_1,G,E) = F ) ) ) ) ) ) ) ) ) ).
fof(t35_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
& v1_hahnban(C,A)
& v4_hahnban(C,A)
& m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),k6_supinf_1)
& v2_hahnban(D,B)
& v3_hahnban(D,B)
& m2_relset_1(D,u1_struct_0(B),k6_supinf_1) )
=> ~ ( ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( E = F
=> r1_xreal_0(k8_funct_2(u1_struct_0(B),k6_supinf_1,D,E),k8_funct_2(u1_struct_0(A),k6_supinf_1,C,F)) ) ) )
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),k6_supinf_1)
& v2_hahnban(E,A)
& v3_hahnban(E,A)
& m2_relset_1(E,u1_struct_0(A),k6_supinf_1) )
=> ~ ( k2_partfun1(u1_struct_0(A),k6_supinf_1,E,u1_struct_0(B)) = D
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k6_supinf_1,E,F),k8_funct_2(u1_struct_0(A),k6_supinf_1,C,F)) ) ) ) ) ) ) ) ) ).
fof(t36_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& l1_normsp_1(A) )
=> ( v1_funct_1(u1_normsp_1(A))
& v1_funct_2(u1_normsp_1(A),u1_struct_0(A),k6_supinf_1)
& v1_hahnban(u1_normsp_1(A),A)
& v6_hahnban(u1_normsp_1(A),A)
& m2_relset_1(u1_normsp_1(A),u1_struct_0(A),k6_supinf_1) ) ) ).
fof(t37_hahnban,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),k6_supinf_1)
& v2_hahnban(C,B)
& v3_hahnban(C,B)
& m2_relset_1(C,u1_struct_0(B),k6_supinf_1) )
=> ~ ( ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( D = E
=> r1_xreal_0(k8_funct_2(u1_struct_0(B),k6_supinf_1,C,D),k1_normsp_1(A,E)) ) ) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k6_supinf_1)
& v2_hahnban(D,A)
& v3_hahnban(D,A)
& m2_relset_1(D,u1_struct_0(A),k6_supinf_1) )
=> ~ ( k2_partfun1(u1_struct_0(A),k6_supinf_1,D,u1_struct_0(B)) = C
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k6_supinf_1,D,E),k1_normsp_1(A,E)) ) ) ) ) ) ) ) ).
fof(t5_hahnban,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_fraenkel(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k3_tarski(A)
=> ( k1_relat_1(B) = k3_tarski(a_1_0_hahnban(A))
& k2_relat_1(B) = k3_tarski(a_1_1_hahnban(A)) ) ) ) ) ).
fof(fraenkel_a_1_0_hahnban,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_fraenkel(B) )
=> ( r2_hidden(A,a_1_0_hahnban(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k1_relat_1(C) ) ) ) ).
fof(fraenkel_a_1_1_hahnban,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_fraenkel(B) )
=> ( r2_hidden(A,a_1_1_hahnban(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k2_relat_1(C) ) ) ) ).
%------------------------------------------------------------------------------