SET007 Axioms: SET007+703.ax
%------------------------------------------------------------------------------
% File : SET007+703 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Cosets in Segre's Product of Partial Linear Spaces
% 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 : pencil_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 39 ( 0 unt; 0 def)
% Number of atoms : 377 ( 30 equ)
% Maximal formula atoms : 23 ( 9 avg)
% Number of connectives : 387 ( 49 ~; 11 |; 176 &)
% ( 5 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 39 ( 38 usr; 0 prp; 1-3 aty)
% Number of functors : 47 ( 47 usr; 9 con; 0-4 aty)
% Number of variables : 150 ( 143 !; 7 ?)
% 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_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(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))
& v1_pencil_2(B,A,A) ) ) ).
fof(d1_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_pencil_2(A,B,C,D) = k8_finseq_1(A,k16_finseq_1(A,B,k5_binarith(C,np__1)),k1_rfinseq(A,B,D)) ) ) ) ).
fof(t1_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_tarski(k2_relat_1(k1_pencil_2(A,B,C,D)),k2_relat_1(B)) ) ) ) ).
fof(t2_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B)) )
=> k3_finseq_1(k1_pencil_2(A,B,C,D)) = k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(B),D),C),np__1) ) ) ) ) ).
fof(t3_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B))
& k3_finseq_1(k1_pencil_2(A,B,C,D)) = np__0 )
=> ( C = np__1
& D = k3_finseq_1(B) ) ) ) ) ) ).
fof(t4_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r1_xreal_0(np__1,E)
& r1_xreal_0(E,k6_xcmplx_0(C,np__1)) )
=> k1_funct_1(k1_pencil_2(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) ).
fof(t5_pencil_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(k3_finseq_1(A),np__1),C)
=> k1_funct_1(k7_finseq_1(A,B),C) = k1_funct_1(B,k6_xcmplx_0(C,k3_finseq_1(A))) ) ) ) ) ).
fof(t6_pencil_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B))
& r1_xreal_0(C,D)
& r1_xreal_0(C,E)
& r1_xreal_0(E,k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(B),D),C),np__1)) )
=> k1_funct_1(k1_pencil_2(A,B,C,D),E) = k1_funct_1(B,k1_nat_1(k1_nat_1(k5_binarith(D,C),E),np__1)) ) ) ) ) ) ).
fof(t7_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m4_pboole(C,A,k12_pralg_1(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k10_pralg_1(A,B,D))))
=> m4_pboole(k2_polynom1(A,C,D,E),A,k12_pralg_1(A,B)) ) ) ) ) ) ).
fof(d2_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pencil_1(A,B))))
=> ( m1_pencil_2(C,A,B)
<=> ? [D] :
( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B))
& C = k4_card_3(D)
& k1_funct_1(D,k3_pencil_1(A,D)) = k2_pre_topc(k1_pencil_1(A,B,k3_pencil_1(A,D))) ) ) ) ) ) ).
fof(t8_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ( r1_tarski(np__2,k1_card_1(k5_subset_1(u1_struct_0(k5_pencil_1(A,B)),C,D)))
=> C = D ) ) ) ) ) ).
fof(d3_pencil_2,axiom,
! [A] :
( l1_pre_topc(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_pencil_2(A,B,C)
<=> ? [D] :
( m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A)))
& B = k1_funct_1(D,np__1)
& C = k1_funct_1(D,k3_finseq_1(D))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,k2_relat_1(D))
=> ( v1_pencil_1(E,A)
& v2_pencil_1(E,A) ) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(k3_finseq_1(D),E)
| r1_tarski(np__2,k1_card_1(k3_xboole_0(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))))) ) ) ) ) ) ) ) ) ).
fof(t9_pencil_2,axiom,
! [A] :
( l1_pre_topc(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_pencil_2(A,B,C)
& ! [D] :
( ( v2_funct_1(D)
& m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B = k1_funct_1(D,np__1)
& C = k1_funct_1(D,k3_finseq_1(D))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,k2_relat_1(D))
=> ( v1_pencil_1(E,A)
& v2_pencil_1(E,A) ) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(k3_finseq_1(D),E)
| r1_tarski(np__2,k1_card_1(k3_xboole_0(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))))) ) ) ) ) ) ) ) ) ) ).
fof(t10_pencil_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_pencil_1(B,A)
& v2_pencil_1(B,A) )
=> r1_pencil_2(A,B,B) ) ) ) ).
fof(t11_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
=> ( ( v1_pencil_1(C,k6_pencil_1(A,B))
& v2_pencil_1(C,k6_pencil_1(A,B))
& v1_pencil_1(D,k6_pencil_1(A,B))
& v2_pencil_1(D,k6_pencil_1(A,B))
& r1_pencil_2(k6_pencil_1(A,B),C,D) )
=> ( v1_realset1(C)
| v1_realset1(D)
| ! [E] :
( ( ~ v13_pencil_1(E)
& v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,B)) )
=> ! [F] :
( ( ~ v13_pencil_1(F)
& v16_pencil_1(F,A)
& m4_pboole(F,A,k12_pralg_1(A,B)) )
=> ( ( C = k4_card_3(E)
& D = k4_card_3(F) )
=> ( k3_pencil_1(A,E) = k3_pencil_1(A,F)
& ! [G] :
( G != k3_pencil_1(A,E)
=> k1_funct_1(E,G) = k1_funct_1(F,G) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_pencil_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_funct_2(C,u1_struct_0(A),u1_struct_0(B))
=> v3_funct_2(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A)) ) ) ) ) ).
fof(d4_pencil_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_pencil_2(C,A,B)
<=> ( v3_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_t_0topsp(C,A,B)
& v3_funct_2(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A))
& v1_t_0topsp(k2_tops_2(A,B,C),B,A) ) ) ) ) ) ).
fof(t13_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v1_funct_1(k2_tops_2(A,A,B))
& v1_funct_2(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(k2_tops_2(A,A,B),A,A)
& m2_relset_1(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(t14_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ v1_realset1(C)
& v1_realset1(k4_pre_topc(A,A,B,C)) ) ) ) ) ).
fof(t15_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ v1_realset1(C)
& v1_realset1(k5_pre_topc(A,A,B,C)) ) ) ) ) ).
fof(t16_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_pencil_1(C,A)
=> v2_pencil_1(k4_pre_topc(A,A,B,C),A) ) ) ) ) ).
fof(t17_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_pencil_1(C,A)
=> v2_pencil_1(k5_pre_topc(A,A,B,C),A) ) ) ) ) ).
fof(t18_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_pencil_1(C,A)
=> v1_pencil_1(k4_pre_topc(A,A,B,C),A) ) ) ) ) ).
fof(t19_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_pencil_1(C,A)
=> v1_pencil_1(k5_pre_topc(A,A,B,C),A) ) ) ) ) ).
fof(t20_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_pencil_2(A,C,D)
=> ( v1_realset1(C)
| v1_realset1(D)
| r1_pencil_2(A,k4_pre_topc(A,A,B,C),k4_pre_topc(A,A,B,D)) ) ) ) ) ) ) ).
fof(t21_pencil_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_pencil_2(A,C,D)
=> ( v1_realset1(C)
| v1_realset1(D)
| r1_pencil_2(A,k5_pre_topc(A,A,B,C),k5_pre_topc(A,A,B,D)) ) ) ) ) ) ) ).
fof(t24_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v10_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(D,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(D,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> m1_pencil_2(k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),D,C),A,B) ) ) ) ) ) ).
fof(t25_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v10_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(D,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(D,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> m1_pencil_2(k5_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),D,C),A,B) ) ) ) ) ) ).
fof(s1_pencil_2,axiom,
( ( f1_s1_pencil_2 = k1_funct_1(f4_s1_pencil_2,np__1)
& f2_s1_pencil_2 = k1_funct_1(f4_s1_pencil_2,k3_finseq_1(f4_s1_pencil_2))
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( r1_xreal_0(np__1,A)
& B = k1_funct_1(f4_s1_pencil_2,A)
& C = k1_funct_1(f4_s1_pencil_2,k1_nat_1(A,np__1)) )
=> ( r1_xreal_0(k3_finseq_1(f4_s1_pencil_2),A)
| p1_s1_pencil_2(B,C) ) ) ) )
=> ? [A] :
( v2_funct_1(A)
& m2_finseq_1(A,f3_s1_pencil_2)
& f1_s1_pencil_2 = k1_funct_1(A,np__1)
& f2_s1_pencil_2 = k1_funct_1(A,k3_finseq_1(A))
& r1_tarski(k2_relat_1(A),k2_relat_1(f4_s1_pencil_2))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(A),B)
| p1_s1_pencil_2(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) ) ).
fof(dt_m1_pencil_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pencil_1(A,B)))) ) ) ).
fof(existence_m1_pencil_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ? [C] : m1_pencil_2(C,A,B) ) ).
fof(dt_k1_pencil_2,axiom,
! [A,B,C,D] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> m2_finseq_1(k1_pencil_2(A,B,C,D),A) ) ).
fof(dt_k2_pencil_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_pre_topc(A)) )
=> m1_subset_1(k2_pencil_2(A,B,C),u1_pre_topc(A)) ) ).
fof(redefinition_k2_pencil_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_pre_topc(A)) )
=> k2_pencil_2(A,B,C) = k9_relat_1(B,C) ) ).
fof(dt_k3_pencil_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_pre_topc(A)) )
=> m1_subset_1(k3_pencil_2(A,B,C),u1_pre_topc(A)) ) ).
fof(redefinition_k3_pencil_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_pencil_2(B,A,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_pre_topc(A)) )
=> k3_pencil_2(A,B,C) = k10_relat_1(B,C) ) ).
fof(t22_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v10_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
=> ( ( v2_pencil_1(C,k6_pencil_1(A,B))
& v1_pencil_1(C,k6_pencil_1(A,B)) )
=> ( v1_realset1(C)
| m1_pencil_2(k3_tarski(a_3_0_pencil_2(A,B,C)),A,B) ) ) ) ) ) ) ).
fof(t23_pencil_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v10_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_pencil_2(C,A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
& ~ v1_realset1(D)
& v2_pencil_1(D,k6_pencil_1(A,B))
& v1_pencil_1(D,k6_pencil_1(A,B))
& C = k3_tarski(a_3_0_pencil_2(A,B,D)) ) ) ) ) ) ).
fof(fraenkel_a_3_0_pencil_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v15_pencil_1(C)
& m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_pencil_1(B,C)))) )
=> ( r2_hidden(A,a_3_0_pencil_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k6_pencil_1(B,C))))
& A = E
& ~ v1_realset1(E)
& v2_pencil_1(E,k6_pencil_1(B,C))
& v1_pencil_1(E,k6_pencil_1(B,C))
& r1_pencil_2(k6_pencil_1(B,C),D,E) ) ) ) ).
%------------------------------------------------------------------------------