SET007 Axioms: SET007+742.ax
%------------------------------------------------------------------------------
% File : SET007+742 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Linear Combinations in 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 51 ( 0 unt; 0 def)
% Number of atoms : 655 ( 45 equ)
% Maximal formula atoms : 19 ( 12 avg)
% Number of connectives : 678 ( 74 ~; 2 |; 415 &)
% ( 11 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 2 con; 0-5 aty)
% Number of variables : 184 ( 167 !; 17 ?)
% 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(t1_rusub_3,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)))
=> ! [C] :
( r1_rlvect_1(k1_rusub_3(A,B),C)
<=> ? [D] :
( m2_rlvect_2(D,A,B)
& C = k13_rlvect_2(A,D) ) ) ) ) ).
fof(t2_rusub_3,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)))
=> ! [C] :
( r2_hidden(C,B)
=> r1_rlvect_1(k1_rusub_3(A,B),C) ) ) ) ).
fof(t3_rusub_3,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) )
=> k1_rusub_3(A,k1_subset_1(u1_struct_0(A))) = k1_rusub_1(A) ) ).
fof(t4_rusub_3,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)))
=> ~ ( k1_rusub_3(A,B) = k1_rusub_1(A)
& B != k1_xboole_0
& B != k7_rlvect_2(A,k1_rlvect_1(A)) ) ) ) ).
fof(t5_rusub_3,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)))
=> ( C = u1_struct_0(B)
=> k1_rusub_3(A,C) = B ) ) ) ) ).
fof(t6_rusub_3,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)
& v1_bhsp_1(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = u1_struct_0(A)
=> k1_rusub_3(A,B) = A ) ) ) ).
fof(t7_rusub_3,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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> m1_rusub_1(k1_rusub_3(A,B),k1_rusub_3(A,C)) ) ) ) ) ).
fof(t8_rusub_3,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)
& v1_bhsp_1(A)
& v2_bhsp_1(A)
& l1_bhsp_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)))
=> ( ( k1_rusub_3(A,B) = A
& r1_tarski(B,C) )
=> k1_rusub_3(A,C) = A ) ) ) ) ).
fof(t9_rusub_3,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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k1_rusub_3(A,k4_subset_1(u1_struct_0(A),B,C)) = k1_rusub_2(A,k1_rusub_3(A,B),k1_rusub_3(A,C)) ) ) ) ).
fof(t10_rusub_3,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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> m1_rusub_1(k1_rusub_3(A,k5_subset_1(u1_struct_0(A),B,C)),k2_rusub_2(A,k1_rusub_3(A,B),k1_rusub_3(A,C))) ) ) ) ).
fof(t11_rusub_3,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_rlvect_3(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(B,C)
& v1_rlvect_3(C,A)
& k1_rusub_3(A,C) = g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) ) ) ) ) ) ).
fof(t12_rusub_3,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)))
=> ~ ( k1_rusub_3(A,B) = A
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(C,B)
& v1_rlvect_3(C,A)
& k1_rusub_3(A,C) = A ) ) ) ) ) ).
fof(d2_rusub_3,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)))
=> ( m1_rusub_3(B,A)
<=> ( v1_rlvect_3(B,A)
& k1_rusub_3(A,B) = g1_bhsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_bhsp_1(A)) ) ) ) ) ).
fof(t13_rusub_3,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)
& v1_bhsp_1(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_rlvect_3(B,A)
& ! [C] :
( m1_rusub_3(C,A)
=> ~ r1_tarski(B,C) ) ) ) ) ).
fof(t14_rusub_3,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)))
=> ~ ( k1_rusub_3(A,B) = A
& ! [C] :
( m1_rusub_3(C,A)
=> ~ r1_tarski(C,B) ) ) ) ) ).
fof(t15_rusub_3,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_rlvect_3(B,A)
& ! [C] :
( m1_rusub_3(C,A)
=> ~ r1_tarski(B,C) ) ) ) ) ).
fof(t16_rusub_3,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_rlvect_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(A))
=> ~ ( r1_tarski(k2_relat_1(D),u1_struct_0(k1_rusub_3(A,C)))
& ! [E] :
( m2_rlvect_2(E,A,C)
=> k9_rlvect_1(A,k12_rlvect_2(A,D,B)) != k13_rlvect_2(A,E) ) ) ) ) ) ) ).
fof(t17_rusub_3,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_rlvect_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(k10_rlvect_2(A,B),u1_struct_0(k1_rusub_3(A,C)))
& ! [D] :
( m2_rlvect_2(D,A,C)
=> k13_rlvect_2(A,B) != k13_rlvect_2(A,D) ) ) ) ) ) ).
fof(t18_rusub_3,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)
=> ! [C] :
( m1_rlvect_2(C,A)
=> ( r1_tarski(k10_rlvect_2(A,C),u1_struct_0(B))
=> ! [D] :
( m1_rlvect_2(D,B)
=> ( D = k7_relat_1(C,u1_struct_0(B))
=> ( k10_rlvect_2(A,C) = k10_rlvect_2(B,D)
& k13_rlvect_2(A,C) = k13_rlvect_2(B,D) ) ) ) ) ) ) ) ).
fof(t19_rusub_3,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)
=> ! [C] :
( m1_rlvect_2(C,B)
=> ? [D] :
( m1_rlvect_2(D,A)
& k10_rlvect_2(B,C) = k10_rlvect_2(A,D)
& k13_rlvect_2(B,C) = k13_rlvect_2(A,D) ) ) ) ) ).
fof(t20_rusub_3,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)
=> ! [C] :
( m1_rlvect_2(C,A)
=> ~ ( r1_tarski(k10_rlvect_2(A,C),u1_struct_0(B))
& ! [D] :
( m1_rlvect_2(D,B)
=> ~ ( k10_rlvect_2(B,D) = k10_rlvect_2(A,C)
& k13_rlvect_2(B,D) = k13_rlvect_2(A,C) ) ) ) ) ) ) ).
fof(t21_rusub_3,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_subset_1(C,u1_struct_0(A))
=> r1_rlvect_1(k1_rusub_3(A,B),C) ) ) ) ).
fof(t22_rusub_3,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)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v1_rlvect_3(C,B)
=> ( v1_rlvect_3(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ) ) ).
fof(t23_rusub_3,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)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_rlvect_3(C,A)
& r1_tarski(C,u1_struct_0(B)) )
=> ( v1_rlvect_3(C,B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) ) ) ) ) ).
fof(t24_rusub_3,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)
=> ! [C] :
( m1_rusub_3(C,B)
=> ? [D] :
( m1_rusub_3(D,A)
& r1_tarski(C,D) ) ) ) ) ).
fof(t25_rusub_3,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_rlvect_3(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( D = k6_subset_1(u1_struct_0(A),B,k7_rlvect_2(A,C))
& r1_rlvect_1(k1_rusub_3(A,D),C) ) ) ) ) ) ) ) ).
fof(t26_rusub_3,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] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( D = k4_subset_1(u1_struct_0(A),B,C)
& v1_rlvect_3(D,A) ) ) ) ) ) ) ).
fof(t27_rusub_3,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)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(C,u1_struct_0(B))
=> m1_rusub_1(k1_rusub_3(A,C),B) ) ) ) ) ).
fof(t28_rusub_3,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)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> k1_rusub_3(A,C) = k1_rusub_3(B,D) ) ) ) ) ) ).
fof(t29_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,B)),C)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& C = k3_rlvect_1(A,B,D) ) ) ) ) ).
fof(t30_rusub_3,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,u1_struct_0(A))
=> r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,B)),B) ) ) ).
fof(t31_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( r2_hidden(D,k3_rusub_1(A,B,k1_rusub_3(A,k7_rlvect_2(A,C))))
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& D = k4_rlvect_1(A,B,k3_rlvect_1(A,C,E)) ) ) ) ) ) ).
fof(t32_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),D)
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& D = k4_rlvect_1(A,k3_rlvect_1(A,B,E),k3_rlvect_1(A,C,F)) ) ) ) ) ) ) ).
fof(t33_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),B)
& r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,B,C)),C) ) ) ) ) ).
fof(t34_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( r2_hidden(E,k3_rusub_1(A,B,k1_rusub_3(A,k8_rlvect_2(A,C,D))))
<=> ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& E = k4_rlvect_1(A,k4_rlvect_1(A,B,k3_rlvect_1(A,C,F)),k3_rlvect_1(A,D,G)) ) ) ) ) ) ) ) ).
fof(t35_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),E)
<=> ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& E = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,F),k3_rlvect_1(A,C,G)),k3_rlvect_1(A,D,H)) ) ) ) ) ) ) ) ) ).
fof(t36_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),B)
& r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),C)
& r1_rlvect_1(k1_rusub_3(A,k9_rlvect_2(A,B,C,D)),D) ) ) ) ) ) ).
fof(t37_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( r2_hidden(F,k3_rusub_1(A,B,k1_rusub_3(A,k9_rlvect_2(A,C,D,E))))
<=> ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& ? [I] :
( m1_subset_1(I,k1_numbers)
& F = k4_rlvect_1(A,k4_rlvect_1(A,k4_rlvect_1(A,B,k3_rlvect_1(A,C,G)),k3_rlvect_1(A,D,H)),k3_rlvect_1(A,E,I)) ) ) ) ) ) ) ) ) ) ).
fof(t38_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_rlvect_1(k1_rusub_3(A,k7_rlvect_2(A,C)),B)
=> ( B = k1_rlvect_1(A)
| k1_rusub_3(A,k7_rlvect_2(A,B)) = k1_rusub_3(A,k7_rlvect_2(A,C)) ) ) ) ) ) ).
fof(t39_rusub_3,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,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( v1_rlvect_3(k8_rlvect_2(A,B,C),A)
& r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,D,E)),B)
& r1_rlvect_1(k1_rusub_3(A,k8_rlvect_2(A,D,E)),C) )
=> ( B = C
| ( k1_rusub_3(A,k8_rlvect_2(A,D,E)) = k1_rusub_3(A,k8_rlvect_2(A,B,C))
& v1_rlvect_3(k8_rlvect_2(A,D,E),A)
& D != E ) ) ) ) ) ) ) ) ).
fof(t40_rusub_3,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] :
( r1_rlvect_1(k1_rusub_1(A),B)
<=> B = k1_rlvect_1(A) ) ) ).
fof(t41_rusub_3,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)
=> ! [C] :
( m1_rusub_1(C,A)
=> ! [D] :
( m1_rusub_1(D,A)
=> ( m1_rusub_1(B,D)
=> m1_rusub_1(k2_rusub_2(A,B,C),D) ) ) ) ) ) ).
fof(t42_rusub_3,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)
=> ! [C] :
( m1_rusub_1(C,A)
=> ! [D] :
( m1_rusub_1(D,A)
=> ( ( m1_rusub_1(B,C)
& m1_rusub_1(B,D) )
=> m1_rusub_1(B,k2_rusub_2(A,C,D)) ) ) ) ) ) ).
fof(t43_rusub_3,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)
=> ! [C] :
( m1_rusub_1(C,A)
=> ! [D] :
( m1_rusub_1(D,A)
=> ( ( m1_rusub_1(B,D)
& m1_rusub_1(C,D) )
=> m1_rusub_1(k1_rusub_2(A,B,C),D) ) ) ) ) ) ).
fof(t44_rusub_3,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)
=> ! [C] :
( m1_rusub_1(C,A)
=> ! [D] :
( m1_rusub_1(D,A)
=> ( m1_rusub_1(B,C)
=> m1_rusub_1(B,k1_rusub_2(A,C,D)) ) ) ) ) ) ).
fof(t45_rusub_3,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)
=> ! [C] :
( m1_rusub_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( m1_rusub_1(B,C)
=> r1_tarski(k3_rusub_1(A,D,B),k3_rusub_1(A,D,C)) ) ) ) ) ) ).
fof(dt_m1_rusub_3,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)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_rusub_3,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) ) ).
fof(dt_k1_rusub_3,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_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_bhsp_1(k1_rusub_3(A,B))
& m1_rusub_1(k1_rusub_3(A,B),A) ) ) ).
fof(d1_rusub_3,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)))
=> ! [C] :
( ( v1_bhsp_1(C)
& m1_rusub_1(C,A) )
=> ( C = k1_rusub_3(A,B)
<=> u1_struct_0(C) = a_2_0_rusub_3(A,B) ) ) ) ) ).
fof(fraenkel_a_2_0_rusub_3,axiom,
! [A,B,C] :
( ( ~ 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_bhsp_1(B)
& l1_bhsp_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_rusub_3(B,C))
<=> ? [D] :
( m2_rlvect_2(D,B,C)
& A = k13_rlvect_2(B,D) ) ) ) ).
%------------------------------------------------------------------------------