SET007 Axioms: SET007+571.ax
%------------------------------------------------------------------------------
% File : SET007+571 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Linear-Metric Space and Isometric Functions
% 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 : vectmetr [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 2 unt; 0 def)
% Number of atoms : 382 ( 43 equ)
% Maximal formula atoms : 25 ( 8 avg)
% Number of connectives : 384 ( 47 ~; 0 |; 221 &)
% ( 11 <=>; 105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 43 ( 41 usr; 1 prp; 0-3 aty)
% Number of functors : 51 ( 51 usr; 4 con; 0-5 aty)
% Number of variables : 124 ( 116 !; 8 ?)
% 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(cc1_vectmetr,axiom,
! [A] :
( l1_metric_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v1_vectmetr(A) )
=> ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v2_vectmetr(A) ) ) ) ).
fof(rc1_vectmetr,axiom,
? [A] :
( l1_metric_1(A)
& ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v1_vectmetr(A)
& v2_vectmetr(A) ) ).
fof(cc2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(B,A) )
=> ( ~ v1_xboole_0(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) ) ).
fof(rc2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(B,A) ) ) ).
fof(fc1_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ~ v1_xboole_0(k1_vectmetr(A)) ) ).
fof(rc3_vectmetr,axiom,
? [A] :
( l1_vectmetr(A)
& v4_vectmetr(A) ) ).
fof(rc4_vectmetr,axiom,
? [A] :
( l1_vectmetr(A)
& ~ v3_struct_0(A)
& v4_vectmetr(A) ) ).
fof(fc2_vectmetr,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_subset_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(k1_numbers,A),A)
& m1_relset_1(E,k2_zfmisc_1(k1_numbers,A),A) )
=> ( ~ v3_struct_0(g1_vectmetr(A,B,C,D,E))
& v4_vectmetr(g1_vectmetr(A,B,C,D,E)) ) ) ).
fof(rc5_vectmetr,axiom,
? [A] :
( l1_vectmetr(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)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v4_vectmetr(A)
& v5_vectmetr(A)
& v6_vectmetr(A) ) ).
fof(fc3_vectmetr,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k5_vectmetr(A))
& v1_group_1(k5_vectmetr(A)) ) ) ).
fof(fc4_vectmetr,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k5_vectmetr(A))
& v1_group_1(k5_vectmetr(A))
& v2_group_1(k5_vectmetr(A))
& v3_group_1(k5_vectmetr(A))
& v4_group_1(k5_vectmetr(A)) ) ) ).
fof(fc5_vectmetr,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_group_2(B,k5_vectmetr(A)) )
=> ( v1_relat_1(k6_vectmetr(A,B))
& v3_relat_2(k6_vectmetr(A,B))
& v8_relat_2(k6_vectmetr(A,B))
& v1_partfun1(k6_vectmetr(A,B),u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A))) ) ) ).
fof(d1_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v1_vectmetr(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)
=> ~ ( r1_xreal_0(np__0,D)
& r1_xreal_0(D,np__1)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( k2_metric_1(A,B,E) = k4_real_1(D,k2_metric_1(A,B,C))
& k2_metric_1(A,E,C) = k4_real_1(k5_real_1(np__1,D),k2_metric_1(A,B,C)) ) ) ) ) ) ) ) ) ).
fof(d2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v2_vectmetr(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)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0)
& ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ~ ( k4_finseq_4(k5_numbers,u1_struct_0(A),F,np__1) = B
& k4_finseq_4(k5_numbers,u1_struct_0(A),F,k3_finseq_1(F)) = C
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,k5_real_1(k3_finseq_1(F),np__1))
& r1_xreal_0(D,k2_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),F,G),k4_finseq_4(k5_numbers,u1_struct_0(A),F,k1_nat_1(G,np__1)))) ) )
& ! [G] :
( m2_finseq_1(G,k1_numbers)
=> ~ ( k3_finseq_1(G) = k5_real_1(k3_finseq_1(F),np__1)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,H)
& r1_xreal_0(H,k3_finseq_1(G)) )
=> k4_finseq_4(k5_numbers,k1_numbers,G,H) = k2_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),F,H),k4_finseq_4(k5_numbers,u1_struct_0(A),F,k1_nat_1(H,np__1))) ) )
& r1_xreal_0(E,k18_complex1(k5_real_1(k2_metric_1(A,B,C),k15_rvsum_1(G)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v1_vectmetr(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)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_finseq_1(E,u1_struct_0(A))
=> ~ ( k4_finseq_4(k5_numbers,u1_struct_0(A),E,np__1) = B
& k4_finseq_4(k5_numbers,u1_struct_0(A),E,k3_finseq_1(E)) = C
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,k5_real_1(k3_finseq_1(E),np__1))
& r1_xreal_0(D,k4_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,F),k4_finseq_4(k5_numbers,u1_struct_0(A),E,k1_nat_1(F,np__1)))) ) )
& ! [F] :
( m2_finseq_1(F,k1_numbers)
=> ( ( k3_finseq_1(F) = k5_real_1(k3_finseq_1(E),np__1)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,k3_finseq_1(F)) )
=> k4_finseq_4(k5_numbers,k1_numbers,F,G) = k4_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,G),k4_finseq_4(k5_numbers,u1_struct_0(A),E,k1_nat_1(G,np__1))) ) ) )
=> k4_metric_1(A,B,C) = k15_rvsum_1(F) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v3_vectmetr(B,A)
<=> ( k2_relat_1(B) = u1_struct_0(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_metric_1(A,C,D) = k2_metric_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) ) ) ) ) ) ).
fof(d4_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( B = k1_vectmetr(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A))
& D = C
& v3_vectmetr(D,A) ) ) ) ) ).
fof(t2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v3_vectmetr(B,A)
=> v2_funct_1(B) ) ) ) ).
fof(t3_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> v3_vectmetr(k2_tops_2(A,A,B),A) ) ) ).
fof(t4_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(C,A)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> v3_vectmetr(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),A) ) ) ) ).
fof(t5_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> v3_vectmetr(k7_grcat_1(A),A) ) ).
fof(d5_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectmetr(A) )
=> ( v5_vectmetr(A)
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_metric_1(A,k3_rlvect_1(A,C,B),k3_rlvect_1(A,D,B)) = k4_real_1(k18_complex1(B),k2_metric_1(A,C,D)) ) ) ) ) ) ).
fof(d6_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectmetr(A) )
=> ( v6_vectmetr(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))
=> k2_metric_1(A,D,C) = k2_metric_1(A,k2_rlvect_1(A,D,B),k2_rlvect_1(A,C,B)) ) ) ) ) ) ).
fof(d7_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_vectmetr(A,B) = k2_metric_1(A,k1_rlvect_1(A),B) ) ) ).
fof(t6_vectmetr,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)
& v5_vectmetr(A)
& l1_vectmetr(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_vectmetr(A,k3_rlvect_1(A,C,B)) = k4_real_1(k18_complex1(B),k3_vectmetr(A,C)) ) ) ) ).
fof(t7_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v9_metric_1(A)
& v6_vectmetr(A)
& l1_vectmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(k3_vectmetr(A,k4_rlvect_1(A,B,C)),k3_real_1(k3_vectmetr(A,B),k3_vectmetr(A,C))) ) ) ) ).
fof(t8_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectmetr(A)
& l1_vectmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_metric_1(A,B,C) = k3_vectmetr(A,k6_rlvect_1(A,C,B)) ) ) ) ).
fof(d8_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& v4_vectmetr(B)
& v5_vectmetr(B)
& v6_vectmetr(B)
& l1_vectmetr(B) )
=> ( B = k4_vectmetr(A)
<=> ( u1_struct_0(B) = k1_euclid(A)
& u1_metric_1(B) = k13_euclid(A)
& u2_struct_0(B) = k5_euclid(A)
& ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k1_binop_1(u1_rlvect_1(B),C,D) = k7_euclid(A,C,D) ) )
& ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k1_binop_1(u2_rlvect_1(B),D,C) = k9_euclid(A,D,C) ) ) ) ) ) ) ).
fof(t9_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A)))
& v3_vectmetr(B,k4_vectmetr(A))
& m2_relset_1(B,u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A))) )
=> k2_relat_1(B) = k1_euclid(A) ) ) ).
fof(d9_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_group_1(B)
& l1_group_1(B) )
=> ( B = k5_vectmetr(A)
<=> ( u1_struct_0(B) = k2_vectmetr(k4_vectmetr(A))
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(C,k2_vectmetr(k4_vectmetr(A)))
& r2_hidden(D,k2_vectmetr(k4_vectmetr(A))) )
=> k1_binop_1(u1_group_1(B),C,D) = k5_relat_1(D,C) ) ) ) ) ) ) ) ).
fof(t10_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_group_1(k5_vectmetr(A)) = k7_grcat_1(k4_vectmetr(A)) ) ).
fof(t11_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_vectmetr(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A)))
& m2_relset_1(C,u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A))) )
=> ( B = C
=> k3_group_1(k5_vectmetr(A),B) = k2_tops_2(k4_vectmetr(A),k4_vectmetr(A),C) ) ) ) ) ).
fof(d10_vectmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_group_2(B,k5_vectmetr(A))
=> ! [C] :
( m2_relset_1(C,u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A)))
=> ( C = k6_vectmetr(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_vectmetr(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_vectmetr(A)))
=> ( r2_hidden(k1_domain_1(u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A)),D,E),C)
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& r2_hidden(F,u1_struct_0(B))
& k1_funct_1(F,D) = E ) ) ) ) ) ) ) ) ).
fof(dt_l1_vectmetr,axiom,
! [A] :
( l1_vectmetr(A)
=> ( l2_rlvect_1(A)
& l1_metric_1(A) ) ) ).
fof(existence_l1_vectmetr,axiom,
? [A] : l1_vectmetr(A) ).
fof(abstractness_v4_vectmetr,axiom,
! [A] :
( l1_vectmetr(A)
=> ( v4_vectmetr(A)
=> A = g1_vectmetr(u1_struct_0(A),u1_metric_1(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)) ) ) ).
fof(dt_k1_vectmetr,axiom,
$true ).
fof(dt_k2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k2_vectmetr(A),k1_zfmisc_1(k1_funct_2(u1_struct_0(A),u1_struct_0(A)))) ) ).
fof(redefinition_k2_vectmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k2_vectmetr(A) = k1_vectmetr(A) ) ).
fof(dt_k3_vectmetr,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectmetr(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_vectmetr(A,B),k1_numbers) ) ).
fof(dt_k4_vectmetr,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k4_vectmetr(A))
& v3_rlvect_1(k4_vectmetr(A))
& v4_rlvect_1(k4_vectmetr(A))
& v5_rlvect_1(k4_vectmetr(A))
& v6_rlvect_1(k4_vectmetr(A))
& v7_rlvect_1(k4_vectmetr(A))
& v6_metric_1(k4_vectmetr(A))
& v7_metric_1(k4_vectmetr(A))
& v8_metric_1(k4_vectmetr(A))
& v9_metric_1(k4_vectmetr(A))
& v4_vectmetr(k4_vectmetr(A))
& v5_vectmetr(k4_vectmetr(A))
& v6_vectmetr(k4_vectmetr(A))
& l1_vectmetr(k4_vectmetr(A)) ) ) ).
fof(dt_k5_vectmetr,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_group_1(k5_vectmetr(A))
& l1_group_1(k5_vectmetr(A)) ) ) ).
fof(dt_k6_vectmetr,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_group_2(B,k5_vectmetr(A)) )
=> m2_relset_1(k6_vectmetr(A,B),u1_struct_0(k4_vectmetr(A)),u1_struct_0(k4_vectmetr(A))) ) ).
fof(dt_g1_vectmetr,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_subset_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(k1_numbers,A),A)
& m1_relset_1(E,k2_zfmisc_1(k1_numbers,A),A) )
=> ( v4_vectmetr(g1_vectmetr(A,B,C,D,E))
& l1_vectmetr(g1_vectmetr(A,B,C,D,E)) ) ) ).
fof(free_g1_vectmetr,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_subset_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(k1_numbers,A),A)
& m1_relset_1(E,k2_zfmisc_1(k1_numbers,A),A) )
=> ! [F,G,H,I,J] :
( g1_vectmetr(A,B,C,D,E) = g1_vectmetr(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
%------------------------------------------------------------------------------