SET007 Axioms: SET007+743.ax
%------------------------------------------------------------------------------
% File : SET007+743 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Dimension of Real Unitary 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 : rusub_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 1 unt; 0 def)
% Number of atoms : 693 ( 43 equ)
% Maximal formula atoms : 20 ( 10 avg)
% Number of connectives : 707 ( 80 ~; 0 |; 459 &)
% ( 14 <=>; 154 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 6 con; 0-5 aty)
% Number of variables : 173 ( 157 !; 16 ?)
% 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_rusub_4,axiom,
? [A] :
( l1_bhsp_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_bhsp_1(A)
& v2_bhsp_1(A)
& v1_rusub_4(A) ) ).
fof(rc2_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ? [B] :
( m1_rusub_1(B,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)
& v1_bhsp_1(B)
& v2_bhsp_1(B)
& v1_rusub_4(B) ) ) ).
fof(cc1_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> v1_rusub_4(B) ) ) ).
fof(rc3_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ? [B] :
( m1_rusub_1(B,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)
& v1_bhsp_1(B)
& v2_bhsp_1(B)
& v1_rusub_4(B) ) ) ).
fof(rc4_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v2_rusub_4(B,A) ) ) ).
fof(rc5_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v1_finset_1(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v2_rusub_4(B,A) ) ) ).
fof(t1_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r1_rlvect_1(k1_rusub_3(A,k2_rlvect_2(A,B,C)),D)
& ~ r1_rlvect_1(k1_rusub_3(A,C),D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,B)
& r1_rlvect_1(k1_rusub_3(A,k2_rlvect_2(A,k4_rlvect_2(A,k2_rlvect_2(A,B,C),k7_rlvect_2(A,E)),k7_rlvect_2(A,D))),E) ) ) ) ) ) ) ) ).
fof(t2_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) = k1_rusub_3(A,B)
& v1_rlvect_3(C,A) )
=> ( r1_xreal_0(k4_card_1(C),k4_card_1(B))
& ? [D] :
( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(D,B)
& k4_card_1(D) = k5_real_1(k4_card_1(B),k4_card_1(C))
& g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) = k1_rusub_3(A,k2_rlvect_2(A,C,D)) ) ) ) ) ) ) ).
fof(d1_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( v1_rusub_4(A)
<=> ? [B] :
( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_rusub_3(B,A) ) ) ) ).
fof(d2_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( v1_rusub_4(A)
<=> ? [B] :
( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_rusub_3(B,A) ) ) ) ).
fof(t3_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( v1_rusub_4(A)
=> ! [B] :
( m1_rusub_3(B,A)
=> v1_finset_1(B) ) ) ) ).
fof(t4_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_rusub_4(A)
& v1_rlvect_3(B,A) )
=> v1_finset_1(B) ) ) ) ).
fof(t5_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_3(B,A)
=> ! [C] :
( m1_rusub_3(C,A)
=> ( v1_rusub_4(A)
=> k1_card_1(B) = k1_card_1(C) ) ) ) ) ).
fof(t6_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> v1_rusub_4(k1_rusub_1(A)) ) ).
fof(t7_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ( v1_rusub_4(A)
=> v1_rusub_4(B) ) ) ) ).
fof(d3_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( v1_rusub_4(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k1_rusub_4(A)
<=> ! [C] :
( m1_rusub_3(C,A)
=> B = k1_card_1(C) ) ) ) ) ) ).
fof(t8_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> r1_xreal_0(k1_rusub_4(B),k1_rusub_4(A)) ) ) ).
fof(t9_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_rlvect_3(B,A)
=> k1_card_1(B) = k1_rusub_4(k1_rusub_3(A,B)) ) ) ) ).
fof(t10_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> k1_rusub_4(A) = k1_rusub_4(k2_rusub_1(A)) ) ).
fof(t11_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ( k1_rusub_4(A) = k1_rusub_4(B)
<=> k2_rusub_1(A) = k2_rusub_1(B) ) ) ) ).
fof(t12_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ( k1_rusub_4(A) = np__0
<=> k2_rusub_1(A) = k1_rusub_1(A) ) ) ).
fof(t13_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ( k1_rusub_4(A) = np__1
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& B != k1_rlvect_1(A)
& k2_rusub_1(A) = k1_rusub_3(A,k7_rlvect_2(A,B)) ) ) ) ).
fof(t14_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ( k1_rusub_4(A) = np__2
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B != C
& v1_rlvect_3(k8_rlvect_2(A,B,C),A)
& k2_rusub_1(A) = k1_rusub_3(A,k8_rlvect_2(A,B,C)) ) ) ) ) ).
fof(t15_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ! [C] :
( m1_rusub_1(C,A)
=> k1_nat_1(k1_rusub_4(k1_rusub_2(A,B,C)),k1_rusub_4(k2_rusub_2(A,B,C))) = k1_nat_1(k1_rusub_4(B),k1_rusub_4(C)) ) ) ) ).
fof(t16_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ! [C] :
( m1_rusub_1(C,A)
=> r1_xreal_0(k5_real_1(k1_nat_1(k1_rusub_4(B),k1_rusub_4(C)),k1_rusub_4(A)),k1_rusub_4(k2_rusub_2(A,B,C))) ) ) ) ).
fof(t17_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ! [C] :
( m1_rusub_1(C,A)
=> ( r1_rusub_2(A,B,C)
=> k1_rusub_4(A) = k1_nat_1(k1_rusub_4(B),k1_rusub_4(C)) ) ) ) ) ).
fof(t18_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k1_rusub_4(A))
<=> ? [D] :
( v1_bhsp_1(D)
& m1_rusub_1(D,A)
& k1_rusub_4(D) = C ) ) ) ) ) ).
fof(d4_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( C = k2_rusub_4(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_bhsp_1(E)
& m1_rusub_1(E,A)
& E = D
& k1_rusub_4(E) = B ) ) ) ) ) ).
fof(t19_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,k1_rusub_4(A))
& v1_xboole_0(k2_rusub_4(A,B)) ) ) ) ).
fof(t20_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,k1_rusub_4(A))
=> k2_rusub_4(A,B) = k1_xboole_0 ) ) ) ).
fof(t21_rusub_4,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_bhsp_1(A)
& v1_rusub_4(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k2_rusub_4(B,C),k2_rusub_4(A,C)) ) ) ) ).
fof(d5_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_rusub_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_rlvect_1(A,k3_rlvect_1(A,C,k5_real_1(np__1,E)),k3_rlvect_1(A,D,E)),B) ) ) ) ) ) ) ) ).
fof(t22_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ( v2_rusub_4(k2_pre_topc(A),A)
& v2_rusub_4(k1_pre_topc(A),A) ) ) ).
fof(t23_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v2_rusub_4(k7_rlvect_2(A,B),A) ) ) ).
fof(d6_rusub_4,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)
=> k3_rusub_4(A,B) = u1_struct_0(B) ) ) ).
fof(d7_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> k4_rusub_4(A,B) = u1_struct_0(B) ) ) ).
fof(t24_rusub_4,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)
=> ( v2_rusub_4(k3_rusub_4(A,B),A)
& r2_hidden(k1_rlvect_1(A),u1_struct_0(B)) ) ) ) ).
fof(t25_rusub_4,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] :
( ( v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(k1_rlvect_1(A),B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(C,B)
=> r2_hidden(k3_rlvect_1(A,C,D),B) ) ) ) ) ) ) ).
fof(d8_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v3_rusub_4(B,A)
<=> ( r2_hidden(u2_struct_0(A),B)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k2_rlvect_1(A,C,D),B)
& r2_hidden(k3_rlvect_1(A,C,E),B) ) ) ) ) ) ) ) ) ) ).
fof(t26_rusub_4,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_xboole_0(B)
& v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(k1_rlvect_1(A),B)
=> ( v3_rusub_4(B,A)
& B = u1_struct_0(k1_rlvect_3(A,B)) ) ) ) ) ).
fof(t27_rusub_4,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)
=> v3_rusub_4(k3_rusub_4(A,B),A) ) ) ).
fof(t28_rusub_4,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] :
( ( v2_rlvect_1(B)
& m1_rlsub_1(B,A) )
=> B = k1_rlvect_3(A,k3_rusub_4(A,B)) ) ) ).
fof(t29_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(k1_rlvect_1(A),B)
=> B = u1_struct_0(k1_rusub_3(A,B)) ) ) ) ).
fof(t30_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_rusub_1(B,A)
=> v3_rusub_4(k4_rusub_4(A,B),A) ) ) ).
fof(t31_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( v1_bhsp_1(B)
& m1_rusub_1(B,A) )
=> B = k1_rusub_3(A,k4_rusub_4(A,B)) ) ) ).
fof(t32_rusub_4,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] :
( ( v2_rlvect_1(B)
& m1_rlsub_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k3_rusub_4(A,B) = C
=> k3_rlsub_1(A,D,B) = k5_rusub_4(A,C,D) ) ) ) ) ) ).
fof(t33_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v2_rusub_4(k5_rusub_4(A,B,C),A) ) ) ) ).
fof(t34_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( v1_bhsp_1(B)
& m1_rusub_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k4_rusub_4(A,B) = C
=> k3_rusub_1(A,D,B) = k5_rusub_4(A,C,D) ) ) ) ) ) ).
fof(t35_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k6_rusub_4(A,C,B) = k6_rusub_4(A,B,C) ) ) ) ).
fof(t36_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k6_rusub_4(A,k7_rlvect_2(A,C),B) = k5_rusub_4(A,B,C) ) ) ) ).
fof(t37_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v2_rusub_4(k7_rusub_4(A,k7_rlvect_2(A,C),B),A) ) ) ) ).
fof(t38_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v2_rusub_4(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v2_rusub_4(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ).
fof(t39_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_rusub_4(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v2_rusub_4(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v2_rusub_4(k7_rusub_4(A,B,C),A) ) ) ) ).
fof(dt_k1_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> m2_subset_1(k1_rusub_4(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_rusub_4,axiom,
$true ).
fof(dt_k3_rusub_4,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)
& l2_rlvect_1(A)
& m1_rlsub_1(B,A) )
=> ( ~ v1_xboole_0(k3_rusub_4(A,B))
& m1_subset_1(k3_rusub_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k4_rusub_4,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_bhsp_1(A)
& l1_bhsp_1(A)
& m1_rusub_1(B,A) )
=> ( ~ v1_xboole_0(k4_rusub_4(A,B))
& m1_subset_1(k4_rusub_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k5_rusub_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k5_rusub_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k6_rusub_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_rusub_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k7_rusub_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k7_rusub_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(commutativity_k7_rusub_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k7_rusub_4(A,B,C) = k7_rusub_4(A,C,B) ) ).
fof(redefinition_k7_rusub_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k7_rusub_4(A,B,C) = k6_rusub_4(A,B,C) ) ).
fof(d9_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_rusub_4(A,B,C) = a_3_0_rusub_4(A,B,C) ) ) ) ).
fof(d10_rusub_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k6_rusub_4(A,B,C) = a_3_1_rusub_4(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_rusub_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_rusub_4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k2_rlvect_1(B,D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_1_rusub_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_1_rusub_4(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(B))
& m1_subset_1(F,u1_struct_0(B))
& A = k2_rlvect_1(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
%------------------------------------------------------------------------------