SET007 Axioms: SET007+806.ax
%------------------------------------------------------------------------------
% File : SET007+806 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Banach Space of Bounded Real Sequences
% 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 : rsspace4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 6 unt; 0 def)
% Number of atoms : 750 ( 65 equ)
% Maximal formula atoms : 30 ( 12 avg)
% Number of connectives : 793 ( 103 ~; 0 |; 511 &)
% ( 16 <=>; 163 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 45 ( 45 usr; 9 con; 0-5 aty)
% Number of variables : 172 ( 168 !; 4 ?)
% 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_rsspace4,axiom,
~ v1_xboole_0(k1_rsspace4) ).
fof(fc2_rsspace4,axiom,
v1_rlsub_1(k1_rsspace4,k7_rsspace) ).
fof(fc3_rsspace4,axiom,
( ~ v3_struct_0(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v2_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v3_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v4_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v5_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v6_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4)))
& v7_rlvect_1(g2_rlvect_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4))) ) ).
fof(fc4_rsspace4,axiom,
( ~ v3_struct_0(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v3_rlvect_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v4_rlvect_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v5_rlvect_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v6_rlvect_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v7_rlvect_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4))
& v1_normsp_1(g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4)) ) ).
fof(fc5_rsspace4,axiom,
( ~ v3_struct_0(k3_rsspace4)
& v3_rlvect_1(k3_rsspace4)
& v4_rlvect_1(k3_rsspace4)
& v5_rlvect_1(k3_rsspace4)
& v6_rlvect_1(k3_rsspace4)
& v7_rlvect_1(k3_rsspace4)
& v2_normsp_1(k3_rsspace4) ) ).
fof(rc1_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ? [C] :
( m1_relset_1(C,A,u1_struct_0(B))
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& v1_partfun1(C,A,u1_struct_0(B))
& v1_rsspace4(C,A,B) ) ) ).
fof(fc6_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ~ v1_xboole_0(k4_rsspace4(A,B)) ) ).
fof(fc7_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v2_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v3_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v4_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v5_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v6_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))))
& v7_rlvect_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)))) ) ) ).
fof(fc8_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k5_rsspace4(A,B))
& v2_rlvect_1(k5_rsspace4(A,B))
& v3_rlvect_1(k5_rsspace4(A,B))
& v4_rlvect_1(k5_rsspace4(A,B))
& v5_rlvect_1(k5_rsspace4(A,B))
& v6_rlvect_1(k5_rsspace4(A,B))
& v7_rlvect_1(k5_rsspace4(A,B)) ) ) ).
fof(fc9_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k9_rsspace4(A,B))
& v3_rlvect_1(k9_rsspace4(A,B))
& v4_rlvect_1(k9_rsspace4(A,B))
& v5_rlvect_1(k9_rsspace4(A,B))
& v6_rlvect_1(k9_rsspace4(A,B))
& v7_rlvect_1(k9_rsspace4(A,B))
& v2_normsp_1(k9_rsspace4(A,B)) ) ) ).
fof(fc10_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_lopban_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k9_rsspace4(A,B))
& v3_rlvect_1(k9_rsspace4(A,B))
& v4_rlvect_1(k9_rsspace4(A,B))
& v5_rlvect_1(k9_rsspace4(A,B))
& v6_rlvect_1(k9_rsspace4(A,B))
& v7_rlvect_1(k9_rsspace4(A,B))
& v2_normsp_1(k9_rsspace4(A,B))
& v4_lopban_1(k9_rsspace4(A,B)) ) ) ).
fof(d1_rsspace4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_rsspace)))
=> ( A = k1_rsspace4
<=> ! [B] :
( r2_hidden(B,A)
<=> ( r2_hidden(B,k1_rsspace)
& v3_seq_2(k2_rsspace(B)) ) ) ) ) ).
fof(d2_rsspace4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k1_rsspace4,k1_numbers)
& m2_relset_1(A,k1_rsspace4,k1_numbers) )
=> ( A = k2_rsspace4
<=> ! [B] :
( r2_hidden(B,k1_rsspace4)
=> k2_seq_1(k1_rsspace4,k1_numbers,A,B) = k3_pscomp_1(k1_pscomp_1(k5_numbers,k1_numbers,k22_seq_1(k2_rsspace(B)))) ) ) ) ).
fof(t1_rsspace4,axiom,
$true ).
fof(t2_rsspace4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v3_seq_2(A)
& k3_pscomp_1(k1_pscomp_1(k5_numbers,k1_numbers,k22_seq_1(A))) = np__0 )
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) ).
fof(d3_rsspace4,axiom,
k3_rsspace4 = g1_normsp_1(k1_rsspace4,k10_rsspace(k7_rsspace,k1_rsspace4),k8_rsspace(k7_rsspace,k1_rsspace4),k9_rsspace(k7_rsspace,k1_rsspace4),k2_rsspace4) ).
fof(t3_rsspace4,axiom,
( u1_struct_0(k3_rsspace4) = k1_rsspace4
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& v3_seq_2(k2_rsspace(A)) ) )
& k1_rlvect_1(k3_rsspace4) = k6_rsspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> A = k2_rsspace(A) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace4))
=> k2_rlvect_1(k3_rsspace4,A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace4))
=> k3_rlvect_1(k3_rsspace4,B,A) = k14_seq_1(k2_rsspace(B),A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> ( k5_rlvect_1(k3_rsspace4,A) = k17_seq_1(k2_rsspace(A))
& k2_rsspace(k5_rlvect_1(k3_rsspace4,A)) = k17_seq_1(k2_rsspace(A)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace4))
=> k6_rlvect_1(k3_rsspace4,A,B) = k10_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> v3_seq_2(k2_rsspace(A)) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> k1_normsp_1(k3_rsspace4,A) = k3_pscomp_1(k1_pscomp_1(k5_numbers,k1_numbers,k22_seq_1(k2_rsspace(A)))) ) ) ).
fof(t4_rsspace4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace4))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace4))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( k1_normsp_1(k3_rsspace4,A) = np__0
=> A = k1_rlvect_1(k3_rsspace4) )
& ( A = k1_rlvect_1(k3_rsspace4)
=> k1_normsp_1(k3_rsspace4,A) = np__0 )
& r1_xreal_0(np__0,k1_normsp_1(k3_rsspace4,A))
& r1_xreal_0(k1_normsp_1(k3_rsspace4,k2_rlvect_1(k3_rsspace4,A,B)),k3_real_1(k1_normsp_1(k3_rsspace4,A),k1_normsp_1(k3_rsspace4,B)))
& k1_normsp_1(k3_rsspace4,k3_rlvect_1(k3_rsspace4,A,C)) = k4_real_1(k18_complex1(C),k1_normsp_1(k3_rsspace4,A)) ) ) ) ) ).
fof(t5_rsspace4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k3_rsspace4))
& m2_relset_1(A,k5_numbers,u1_struct_0(k3_rsspace4)) )
=> ( v1_rsspace3(A,k3_rsspace4)
=> v4_normsp_1(A,k3_rsspace4) ) ) ).
fof(d4_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& m2_relset_1(C,A,u1_struct_0(B)) )
=> ( v1_rsspace4(C,A,B)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& r1_xreal_0(np__0,D)
& ! [E] :
( m1_subset_1(E,A)
=> r1_xreal_0(k1_normsp_1(B,k8_funct_2(A,u1_struct_0(B),C,E)),D) ) ) ) ) ) ) ).
fof(t6_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& m2_relset_1(C,A,u1_struct_0(B)) )
=> ( ! [D] :
( m1_subset_1(D,A)
=> k8_funct_2(A,u1_struct_0(B),C,D) = k1_rlvect_1(B) )
=> v1_rsspace4(C,A,B) ) ) ) ) ).
fof(d5_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_lopban_1(A,B))))
=> ( C = k4_rsspace4(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(B))
& v1_rsspace4(D,A,B)
& m2_relset_1(D,A,u1_struct_0(B)) ) ) ) ) ) ) ).
fof(t7_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> v1_rlsub_1(k4_rsspace4(A,B),k5_lopban_1(A,B)) ) ) ).
fof(t8_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> m1_rlsub_1(g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))),k5_lopban_1(A,B)) ) ) ).
fof(d6_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> k5_rsspace4(A,B) = g2_rlvect_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B))) ) ) ).
fof(t9_rsspace4,axiom,
$true ).
fof(t10_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_rsspace4(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_rsspace4(A,B)))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,u1_struct_0(B))
& v1_rsspace4(F,A,B)
& m2_relset_1(F,A,u1_struct_0(B)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,A,u1_struct_0(B))
& v1_rsspace4(G,A,B)
& m2_relset_1(G,A,u1_struct_0(B)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,A,u1_struct_0(B))
& v1_rsspace4(H,A,B)
& m2_relset_1(H,A,u1_struct_0(B)) )
=> ( ( F = C
& G = D
& H = E )
=> ( E = k4_rlvect_1(k5_rsspace4(A,B),C,D)
<=> ! [I] :
( m1_subset_1(I,A)
=> k8_funct_2(A,u1_struct_0(B),H,I) = k4_rlvect_1(B,k8_funct_2(A,u1_struct_0(B),F,I),k8_funct_2(A,u1_struct_0(B),G,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_rsspace4(A,B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& v1_rsspace4(E,A,B)
& m2_relset_1(E,A,u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,u1_struct_0(B))
& v1_rsspace4(F,A,B)
& m2_relset_1(F,A,u1_struct_0(B)) )
=> ( ( E = C
& F = D )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( D = k3_rlvect_1(k5_rsspace4(A,B),C,G)
<=> ! [H] :
( m1_subset_1(H,A)
=> k8_funct_2(A,u1_struct_0(B),F,H) = k3_rlvect_1(B,k8_funct_2(A,u1_struct_0(B),E,H),G) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> k1_rlvect_1(k5_rsspace4(A,B)) = k2_funcop_1(A,k1_rlvect_1(B)) ) ) ).
fof(d7_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( r2_hidden(C,k4_rsspace4(A,B))
=> k6_rsspace4(A,B,C) = C ) ) ) ).
fof(t13_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& v1_rsspace4(C,A,B)
& m2_relset_1(C,A,u1_struct_0(B)) )
=> ( ~ v1_xboole_0(k7_rsspace4(A,B,C))
& v1_seq_4(k7_rsspace4(A,B,C)) ) ) ) ) ).
fof(t14_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& m2_relset_1(C,A,u1_struct_0(B)) )
=> ( v1_rsspace4(C,A,B)
<=> v1_seq_4(k7_rsspace4(A,B,C)) ) ) ) ) ).
fof(t15_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k4_rsspace4(A,B),k1_numbers)
& m2_relset_1(C,k4_rsspace4(A,B),k1_numbers)
& ! [D] :
( r2_hidden(D,k4_rsspace4(A,B))
=> k2_seq_1(k4_rsspace4(A,B),k1_numbers,C,D) = k3_pscomp_1(k7_rsspace4(A,B,k6_rsspace4(A,B,D))) ) ) ) ) ).
fof(d9_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_rsspace4(A,B),k1_numbers)
& m2_relset_1(C,k4_rsspace4(A,B),k1_numbers) )
=> ( C = k8_rsspace4(A,B)
<=> ! [D] :
( r2_hidden(D,k4_rsspace4(A,B))
=> k2_seq_1(k4_rsspace4(A,B),k1_numbers,C,D) = k3_pscomp_1(k7_rsspace4(A,B,k6_rsspace4(A,B,D))) ) ) ) ) ) ).
fof(t16_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& v1_rsspace4(C,A,B)
& m2_relset_1(C,A,u1_struct_0(B)) )
=> k6_rsspace4(A,B,C) = C ) ) ) ).
fof(t17_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& v1_rsspace4(C,A,B)
& m2_relset_1(C,A,u1_struct_0(B)) )
=> k2_seq_1(k4_rsspace4(A,B),k1_numbers,k8_rsspace4(A,B),C) = k3_pscomp_1(k7_rsspace4(A,B,C)) ) ) ) ).
fof(d10_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> k9_rsspace4(A,B) = g1_normsp_1(k4_rsspace4(A,B),k10_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k9_rsspace(k5_lopban_1(A,B),k4_rsspace4(A,B)),k8_rsspace4(A,B)) ) ) ).
fof(t18_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> k2_funcop_1(A,k1_rlvect_1(B)) = k1_rlvect_1(k9_rsspace4(A,B)) ) ) ).
fof(t19_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(B))
& v1_rsspace4(D,A,B)
& m2_relset_1(D,A,u1_struct_0(B)) )
=> ( D = C
=> ! [E] :
( m1_subset_1(E,A)
=> r1_xreal_0(k1_normsp_1(B,k8_funct_2(A,u1_struct_0(B),D,E)),k1_normsp_1(k9_rsspace4(A,B),C)) ) ) ) ) ) ) ).
fof(t20_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> r1_xreal_0(np__0,k1_normsp_1(k9_rsspace4(A,B),C)) ) ) ) ).
fof(t21_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ( C = k1_rlvect_1(k9_rsspace4(A,B))
=> np__0 = k1_normsp_1(k9_rsspace4(A,B),C) ) ) ) ) ).
fof(t22_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_rsspace4(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k9_rsspace4(A,B)))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,u1_struct_0(B))
& v1_rsspace4(F,A,B)
& m2_relset_1(F,A,u1_struct_0(B)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,A,u1_struct_0(B))
& v1_rsspace4(G,A,B)
& m2_relset_1(G,A,u1_struct_0(B)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,A,u1_struct_0(B))
& v1_rsspace4(H,A,B)
& m2_relset_1(H,A,u1_struct_0(B)) )
=> ( ( F = C
& G = D
& H = E )
=> ( E = k2_rlvect_1(k9_rsspace4(A,B),C,D)
<=> ! [I] :
( m1_subset_1(I,A)
=> k8_funct_2(A,u1_struct_0(B),H,I) = k4_rlvect_1(B,k8_funct_2(A,u1_struct_0(B),F,I),k8_funct_2(A,u1_struct_0(B),G,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_rsspace4(A,B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& v1_rsspace4(E,A,B)
& m2_relset_1(E,A,u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,u1_struct_0(B))
& v1_rsspace4(F,A,B)
& m2_relset_1(F,A,u1_struct_0(B)) )
=> ( ( E = C
& F = D )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( D = k3_rlvect_1(k9_rsspace4(A,B),C,G)
<=> ! [H] :
( m1_subset_1(H,A)
=> k8_funct_2(A,u1_struct_0(B),F,H) = k3_rlvect_1(B,k8_funct_2(A,u1_struct_0(B),E,H),G) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_rsspace4(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( k1_normsp_1(k9_rsspace4(A,B),C) = np__0
=> C = k1_rlvect_1(k9_rsspace4(A,B)) )
& ( C = k1_rlvect_1(k9_rsspace4(A,B))
=> k1_normsp_1(k9_rsspace4(A,B),C) = np__0 )
& k1_normsp_1(k9_rsspace4(A,B),k3_rlvect_1(k9_rsspace4(A,B),C,E)) = k4_real_1(k18_complex1(E),k1_normsp_1(k9_rsspace4(A,B),C))
& r1_xreal_0(k1_normsp_1(k9_rsspace4(A,B),k2_rlvect_1(k9_rsspace4(A,B),C,D)),k3_real_1(k1_normsp_1(k9_rsspace4(A,B),C),k1_normsp_1(k9_rsspace4(A,B),D))) ) ) ) ) ) ) ).
fof(t25_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> v2_normsp_1(k9_rsspace4(A,B)) ) ) ).
fof(t26_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k9_rsspace4(A,B))
& v3_rlvect_1(k9_rsspace4(A,B))
& v4_rlvect_1(k9_rsspace4(A,B))
& v5_rlvect_1(k9_rsspace4(A,B))
& v6_rlvect_1(k9_rsspace4(A,B))
& v7_rlvect_1(k9_rsspace4(A,B))
& v2_normsp_1(k9_rsspace4(A,B))
& l1_normsp_1(k9_rsspace4(A,B)) ) ) ) ).
fof(t27_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_rsspace4(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_rsspace4(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k9_rsspace4(A,B)))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,u1_struct_0(B))
& v1_rsspace4(F,A,B)
& m2_relset_1(F,A,u1_struct_0(B)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,A,u1_struct_0(B))
& v1_rsspace4(G,A,B)
& m2_relset_1(G,A,u1_struct_0(B)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,A,u1_struct_0(B))
& v1_rsspace4(H,A,B)
& m2_relset_1(H,A,u1_struct_0(B)) )
=> ( ( F = C
& G = D
& H = E )
=> ( E = k6_rlvect_1(k9_rsspace4(A,B),C,D)
<=> ! [I] :
( m1_subset_1(I,A)
=> k8_funct_2(A,u1_struct_0(B),H,I) = k6_rlvect_1(B,k8_funct_2(A,u1_struct_0(B),F,I),k8_funct_2(A,u1_struct_0(B),G,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t28_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( v4_lopban_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k9_rsspace4(A,B)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k9_rsspace4(A,B))) )
=> ( v1_rsspace3(C,k9_rsspace4(A,B))
=> v4_normsp_1(C,k9_rsspace4(A,B)) ) ) ) ) ) ).
fof(t29_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_lopban_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k9_rsspace4(A,B))
& v3_rlvect_1(k9_rsspace4(A,B))
& v4_rlvect_1(k9_rsspace4(A,B))
& v5_rlvect_1(k9_rsspace4(A,B))
& v6_rlvect_1(k9_rsspace4(A,B))
& v7_rlvect_1(k9_rsspace4(A,B))
& v2_normsp_1(k9_rsspace4(A,B))
& v4_lopban_1(k9_rsspace4(A,B))
& l1_normsp_1(k9_rsspace4(A,B)) ) ) ) ).
fof(dt_k1_rsspace4,axiom,
m1_subset_1(k1_rsspace4,k1_zfmisc_1(u1_struct_0(k7_rsspace))) ).
fof(dt_k2_rsspace4,axiom,
( v1_funct_1(k2_rsspace4)
& v1_funct_2(k2_rsspace4,k1_rsspace4,k1_numbers)
& m2_relset_1(k2_rsspace4,k1_rsspace4,k1_numbers) ) ).
fof(dt_k3_rsspace4,axiom,
( ~ v3_struct_0(k3_rsspace4)
& l1_normsp_1(k3_rsspace4) ) ).
fof(dt_k4_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> m1_subset_1(k4_rsspace4(A,B),k1_zfmisc_1(u1_struct_0(k5_lopban_1(A,B)))) ) ).
fof(dt_k5_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k5_rsspace4(A,B))
& v3_rlvect_1(k5_rsspace4(A,B))
& v4_rlvect_1(k5_rsspace4(A,B))
& v5_rlvect_1(k5_rsspace4(A,B))
& v6_rlvect_1(k5_rsspace4(A,B))
& v7_rlvect_1(k5_rsspace4(A,B))
& l2_rlvect_1(k5_rsspace4(A,B)) ) ) ).
fof(dt_k6_rsspace4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( v1_funct_1(k6_rsspace4(A,B,C))
& v1_funct_2(k6_rsspace4(A,B,C),A,u1_struct_0(B))
& v1_rsspace4(k6_rsspace4(A,B,C),A,B)
& m2_relset_1(k6_rsspace4(A,B,C),A,u1_struct_0(B)) ) ) ).
fof(dt_k7_rsspace4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& m1_relset_1(C,A,u1_struct_0(B)) )
=> ( ~ v1_xboole_0(k7_rsspace4(A,B,C))
& m1_subset_1(k7_rsspace4(A,B,C),k1_zfmisc_1(k1_numbers)) ) ) ).
fof(dt_k8_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( v1_funct_1(k8_rsspace4(A,B))
& v1_funct_2(k8_rsspace4(A,B),k4_rsspace4(A,B),k1_numbers)
& m2_relset_1(k8_rsspace4(A,B),k4_rsspace4(A,B),k1_numbers) ) ) ).
fof(dt_k9_rsspace4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ( ~ v3_struct_0(k9_rsspace4(A,B))
& l1_normsp_1(k9_rsspace4(A,B)) ) ) ).
fof(d8_rsspace4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,u1_struct_0(B))
& m2_relset_1(C,A,u1_struct_0(B)) )
=> k7_rsspace4(A,B,C) = a_3_0_rsspace4(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_rsspace4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C)
& v1_funct_1(D)
& v1_funct_2(D,B,u1_struct_0(C))
& m2_relset_1(D,B,u1_struct_0(C)) )
=> ( r2_hidden(A,a_3_0_rsspace4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = k1_normsp_1(C,k8_funct_2(B,u1_struct_0(C),D,E)) ) ) ) ).
%------------------------------------------------------------------------------