SET007 Axioms: SET007+257.ax
%------------------------------------------------------------------------------
% File : SET007+257 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Normed 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 : normsp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 75 ( 17 unt; 0 def)
% Number of atoms : 695 ( 53 equ)
% Maximal formula atoms : 20 ( 9 avg)
% Number of connectives : 682 ( 62 ~; 0 |; 438 &)
% ( 13 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 4 con; 0-5 aty)
% Number of variables : 193 ( 180 !; 13 ?)
% 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_normsp_1,axiom,
? [A] :
( l1_normsp_1(A)
& v1_normsp_1(A) ) ).
fof(rc2_normsp_1,axiom,
? [A] :
( l1_normsp_1(A)
& ~ v3_struct_0(A) ) ).
fof(rc3_normsp_1,axiom,
? [A] :
( l1_normsp_1(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)
& v1_normsp_1(A)
& v2_normsp_1(A) ) ).
fof(d1_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_normsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_normsp_1(A,B) = k2_seq_1(u1_struct_0(A),k1_numbers,u1_normsp_1(A),B) ) ) ).
fof(d2_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_normsp_1(A) )
=> ( v2_normsp_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,k1_numbers)
=> ( ( k1_normsp_1(A,B) = np__0
=> B = k1_rlvect_1(A) )
& ( B = k1_rlvect_1(A)
=> k1_normsp_1(A,B) = np__0 )
& k1_normsp_1(A,k3_rlvect_1(A,B,D)) = k4_real_1(k18_complex1(D),k1_normsp_1(A,B))
& r1_xreal_0(k1_normsp_1(A,k2_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) ) ) ) ).
fof(t1_normsp_1,axiom,
$true ).
fof(t2_normsp_1,axiom,
$true ).
fof(t3_normsp_1,axiom,
$true ).
fof(t4_normsp_1,axiom,
$true ).
fof(t5_normsp_1,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) )
=> k1_normsp_1(A,k1_rlvect_1(A)) = np__0 ) ).
fof(t6_normsp_1,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_subset_1(B,u1_struct_0(A))
=> k1_normsp_1(A,k5_rlvect_1(A,B)) = k1_normsp_1(A,B) ) ) ).
fof(t7_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) ).
fof(t8_normsp_1,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_subset_1(B,u1_struct_0(A))
=> r1_xreal_0(np__0,k1_normsp_1(A,B)) ) ) ).
fof(t9_normsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( ~ 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) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> r1_xreal_0(k1_normsp_1(C,k4_rlvect_1(C,k3_rlvect_1(C,D,A),k3_rlvect_1(C,E,B))),k3_real_1(k4_real_1(k18_complex1(A),k1_normsp_1(C,D)),k4_real_1(k18_complex1(B),k1_normsp_1(C,E)))) ) ) ) ) ) ).
fof(t10_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_normsp_1(A,k6_rlvect_1(A,B,C)) = np__0
<=> B = C ) ) ) ) ).
fof(t11_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_normsp_1(A,k6_rlvect_1(A,B,C)) = k1_normsp_1(A,k6_rlvect_1(A,C,B)) ) ) ) ).
fof(t12_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C)),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) ).
fof(t13_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(k18_complex1(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) ).
fof(t14_normsp_1,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_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))
=> r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,k6_rlvect_1(A,B,D)),k1_normsp_1(A,k6_rlvect_1(A,D,C)))) ) ) ) ) ).
fof(t15_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& k1_normsp_1(A,k6_rlvect_1(A,B,C)) = np__0 ) ) ) ) ).
fof(t16_normsp_1,axiom,
$true ).
fof(t17_normsp_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(B))
& m2_relset_1(A,k5_numbers,u1_struct_0(B)) )
<=> ( k1_relat_1(A) = k5_numbers
& ! [D] :
( r2_hidden(D,k5_numbers)
=> m1_subset_1(k1_funct_1(A,D),u1_struct_0(B)) ) ) ) ) ) ) ).
fof(t18_normsp_1,axiom,
$true ).
fof(t19_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A))
& k2_relat_1(C) = k1_struct_0(A,B) ) ) ) ).
fof(t20_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ~ ( ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_relat_1(B) != k1_struct_0(A,C) ) ) ) ) ).
fof(t21_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_relat_1(B) = k1_struct_0(A,C) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) ) ) ) ) ).
fof(t22_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) ) ) ) ) ).
fof(t23_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) ) ) ) ) ).
fof(t24_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ~ ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) ).
fof(t25_normsp_1,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] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A))
& k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ).
fof(d3_normsp_1,axiom,
$true ).
fof(d4_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v5_seqm_3(B)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) ).
fof(t26_normsp_1,axiom,
$true ).
fof(t27_normsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v5_seqm_3(B)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_relat_1(B) = k1_struct_0(A,C) ) ) ) ) ).
fof(d5_normsp_1,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,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k3_normsp_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ).
fof(d6_normsp_1,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,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k4_normsp_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ).
fof(d7_normsp_1,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,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k5_normsp_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ).
fof(d8_normsp_1,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,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k6_normsp_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k3_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ).
fof(d9_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ).
fof(t28_normsp_1,axiom,
$true ).
fof(t29_normsp_1,axiom,
$true ).
fof(t30_normsp_1,axiom,
$true ).
fof(t31_normsp_1,axiom,
$true ).
fof(t32_normsp_1,axiom,
$true ).
fof(t33_normsp_1,axiom,
$true ).
fof(t34_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> v4_normsp_1(k3_normsp_1(A,B,C),A) ) ) ) ) ).
fof(t35_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> v4_normsp_1(k4_normsp_1(A,B,C),A) ) ) ) ) ).
fof(t36_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> v4_normsp_1(k5_normsp_1(A,C,B),A) ) ) ) ) ).
fof(t37_normsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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,k5_numbers,u1_struct_0(B))
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( v4_normsp_1(C,B)
=> v4_normsp_1(k6_normsp_1(B,C,A),B) ) ) ) ) ).
fof(d10_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k7_normsp_1(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k1_normsp_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) ).
fof(t38_normsp_1,axiom,
$true ).
fof(t39_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(B,A)
=> v4_seq_2(k7_normsp_1(A,B)) ) ) ) ).
fof(d11_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k8_normsp_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ) ).
fof(t40_normsp_1,axiom,
$true ).
fof(t41_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(C,A)
& k8_normsp_1(A,C) = B )
=> ( v4_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B)))
& k2_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B))) = np__0 ) ) ) ) ) ).
fof(t42_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> k8_normsp_1(A,k3_normsp_1(A,B,C)) = k4_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) ).
fof(t43_normsp_1,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> k8_normsp_1(A,k4_normsp_1(A,B,C)) = k6_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) ).
fof(t44_normsp_1,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_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> k8_normsp_1(A,k5_normsp_1(A,C,B)) = k6_rlvect_1(A,k8_normsp_1(A,C),B) ) ) ) ) ).
fof(t45_normsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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,k5_numbers,u1_struct_0(B))
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( v4_normsp_1(C,B)
=> k8_normsp_1(B,k6_normsp_1(B,C,A)) = k3_rlvect_1(B,k8_normsp_1(B,C),A) ) ) ) ) ).
fof(dt_l1_normsp_1,axiom,
! [A] :
( l1_normsp_1(A)
=> l2_rlvect_1(A) ) ).
fof(existence_l1_normsp_1,axiom,
? [A] : l1_normsp_1(A) ).
fof(abstractness_v1_normsp_1,axiom,
! [A] :
( l1_normsp_1(A)
=> ( v1_normsp_1(A)
=> A = g1_normsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_normsp_1(A)) ) ) ).
fof(redefinition_v3_normsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v3_normsp_1(B,A)
<=> v5_seqm_3(B) ) ) ).
fof(dt_k1_normsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_normsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_normsp_1(A,B),k1_numbers) ) ).
fof(dt_k2_normsp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ).
fof(redefinition_k2_normsp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k3_normsp_1,axiom,
! [A,B,C] :
( ( ~ 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)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k3_normsp_1(A,B,C))
& v1_funct_2(k3_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k3_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k4_normsp_1,axiom,
! [A,B,C] :
( ( ~ 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)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k4_normsp_1(A,B,C))
& v1_funct_2(k4_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k4_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k5_normsp_1,axiom,
! [A,B,C] :
( ( ~ 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)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_funct_1(k5_normsp_1(A,B,C))
& v1_funct_2(k5_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k5_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k6_normsp_1,axiom,
! [A,B,C] :
( ( ~ 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)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,k1_numbers) )
=> ( v1_funct_1(k6_normsp_1(A,B,C))
& v1_funct_2(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k7_normsp_1,axiom,
! [A,B] :
( ( ~ 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(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k7_normsp_1(A,B))
& v1_funct_2(k7_normsp_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k7_normsp_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k8_normsp_1,axiom,
! [A,B] :
( ( ~ 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(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m1_subset_1(k8_normsp_1(A,B),u1_struct_0(A)) ) ).
fof(dt_u1_normsp_1,axiom,
! [A] :
( l1_normsp_1(A)
=> ( v1_funct_1(u1_normsp_1(A))
& v1_funct_2(u1_normsp_1(A),u1_struct_0(A),k1_numbers)
& m2_relset_1(u1_normsp_1(A),u1_struct_0(A),k1_numbers) ) ) ).
fof(dt_g1_normsp_1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,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(k1_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
& v1_funct_1(E)
& v1_funct_2(E,A,k1_numbers)
& m1_relset_1(E,A,k1_numbers) )
=> ( v1_normsp_1(g1_normsp_1(A,B,C,D,E))
& l1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) ).
fof(free_g1_normsp_1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,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(k1_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
& v1_funct_1(E)
& v1_funct_2(E,A,k1_numbers)
& m1_relset_1(E,A,k1_numbers) )
=> ! [F,G,H,I,J] :
( g1_normsp_1(A,B,C,D,E) = g1_normsp_1(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
%------------------------------------------------------------------------------