SET007 Axioms: SET007+877.ax
%------------------------------------------------------------------------------
% File : SET007+877 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Characterization of Collineations of the Segre Product
% Version : [Urb08] axioms.
% English : On the Characterization of Collineations of the Segre Product of
% Strongly Connected Partial Linear Spaces
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 38 ( 0 unt; 0 def)
% Number of atoms : 555 ( 74 equ)
% Maximal formula atoms : 35 ( 14 avg)
% Number of connectives : 607 ( 90 ~; 3 |; 263 &)
% ( 10 <=>; 241 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 16 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 41 ( 40 usr; 0 prp; 1-4 aty)
% Number of functors : 31 ( 31 usr; 3 con; 0-4 aty)
% Number of variables : 215 ( 206 !; 9 ?)
% 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_pencil_3,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_pencil_1(A,C,D)
=> r1_pencil_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(t2_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v2_pralg_1(C)
& v14_pencil_1(C)
& m1_pboole(C,A) )
=> ~ v3_realset2(k10_pralg_1(A,C,B)) ) ) ) ).
fof(t3_pencil_3,axiom,
! [A] :
( ( ~ v3_pencil_1(A)
& v6_pencil_1(A)
& l1_pre_topc(A) )
=> ( v10_pencil_1(A)
=> v8_pencil_1(A) ) ) ).
fof(t4_pencil_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& v6_pencil_1(A)
& l1_pre_topc(A) )
=> ( v10_pencil_1(A)
=> v9_pencil_1(A) ) ) ).
fof(t5_pencil_3,axiom,
! [A] :
( ( ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_pre_topc(A))
=> ~ ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(C,B) ) ) ) ).
fof(t6_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_pencil_1(A,B)))
=> m2_pboole(C,A,k12_pralg_1(A,B)) ) ) ) ).
fof(t7_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> k1_funct_1(k12_pralg_1(A,B),C) = k2_pre_topc(k10_pralg_1(A,B,C)) ) ) ) ).
fof(t8_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v11_pencil_1(C)
& v14_pencil_1(C)
& m1_pboole(C,A) )
=> ? [D] :
( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,C))
& k3_pencil_1(A,D) = B
& m1_pencil_2(k4_card_3(D),A,C) ) ) ) ) ).
fof(t9_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v11_pencil_1(C)
& v14_pencil_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_pencil_1(A,C)))
=> ? [E] :
( ~ v13_pencil_1(E)
& v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,C))
& k3_pencil_1(A,E) = B
& m1_pencil_2(k4_card_3(E),A,C)
& r2_hidden(D,k4_card_3(E)) ) ) ) ) ) ).
fof(t10_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v13_pencil_1(C)
& v16_pencil_1(C,A)
& m4_pboole(C,A,k12_pralg_1(A,B)) )
=> ( m1_pencil_2(k4_card_3(C),A,B)
=> k1_funct_1(C,k3_pencil_1(A,C)) = k2_pre_topc(k1_pencil_1(A,B,k3_pencil_1(A,C))) ) ) ) ) ).
fof(t11_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v11_pencil_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v13_pencil_1(C)
& v16_pencil_1(C,A)
& m4_pboole(C,A,k12_pralg_1(A,B)) )
=> ! [D] :
( ( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B)) )
=> ( ( m1_pencil_2(k4_card_3(C),A,B)
& m1_pencil_2(k4_card_3(D),A,B)
& k3_pencil_1(A,C) = k3_pencil_1(A,D) )
=> ( r2_subset_1(k4_card_3(C),k4_card_3(D))
| r6_pboole(A,C,D) ) ) ) ) ) ) ).
fof(t12_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v13_pencil_1(C)
& v16_pencil_1(C,A)
& m4_pboole(C,A,k12_pralg_1(A,B)) )
=> ! [D] :
( m1_subset_1(D,u1_pre_topc(k2_pencil_1(A,B,k3_pencil_1(A,C))))
=> m1_subset_1(k4_card_3(k2_polynom1(A,C,k3_pencil_1(A,C),D)),u1_pre_topc(k6_pencil_1(A,B))) ) ) ) ) ).
fof(t13_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C)))
=> ! [E] :
( ( ~ v13_pencil_1(E)
& v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,B)) )
=> ( C != k3_pencil_1(A,E)
=> ( ~ v13_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)))
& v16_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A)
& m4_pboole(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A,k12_pralg_1(A,B)) ) ) ) ) ) ) ) ).
fof(t14_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k2_pencil_1(A,B,C))))
=> ! [E] :
( ( ~ v13_pencil_1(E)
& v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,B)) )
=> m1_subset_1(k4_card_3(k2_polynom1(A,E,C,D)),k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) ) ) ) ) ) ).
fof(t15_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( ~ v1_xboole_0(k1_funct_1(k1_pzfmisc1(A,B),C))
& v1_realset1(k1_funct_1(k1_pzfmisc1(A,B),C)) ) ) ) ) ).
fof(t16_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v15_pencil_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_pre_topc(k2_pencil_1(A,C,B)))
=> ! [E] :
( m2_pboole(E,A,k12_pralg_1(A,C))
=> m1_subset_1(k4_card_3(k2_polynom1(A,k1_pzfmisc1(A,E),B,D)),u1_pre_topc(k6_pencil_1(A,C))) ) ) ) ) ) ).
fof(t17_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_pencil_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k6_pencil_1(A,B)))
=> ( C != D
=> ( r1_pencil_1(k6_pencil_1(A,B),C,D)
<=> ! [E] :
( m1_pboole(E,A)
=> ! [F] :
( m1_pboole(F,A)
=> ~ ( E = C
& F = D
& ! [G] :
( m1_subset_1(G,A)
=> ~ ( ! [H] :
( m1_subset_1(H,u1_struct_0(k2_pencil_1(A,B,G)))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k2_pencil_1(A,B,G)))
=> ( ( H = k1_funct_1(E,G)
& I = k1_funct_1(F,G) )
=> ( H != I
& r1_pencil_1(k2_pencil_1(A,B,G),H,I) ) ) ) )
& ! [H] :
( m1_subset_1(H,A)
=> ( H != G
=> k1_funct_1(E,H) = k1_funct_1(F,H) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v13_pencil_1(C)
& v16_pencil_1(C,A)
& m4_pboole(C,A,k12_pralg_1(A,B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,C))))
=> ? [E] :
( m1_pboole(E,A)
& r2_hidden(E,k4_card_3(C))
& k1_tarski(k2_polynom1(A,E,k3_pencil_1(A,C),D)) = k4_card_3(k2_polynom1(A,C,k3_pencil_1(A,C),k1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,C)),D))) ) ) ) ) ) ).
fof(t19_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( k1_funct_1(B,D) != k1_funct_1(C,D)
=> k1_pencil_3(A,B,C) = k1_nat_1(k1_pencil_3(A,B,k2_polynom1(A,C,D,k1_funct_1(B,D))),np__1) ) ) ) ) ) ).
fof(d2_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ( r1_pencil_3(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k6_pencil_1(A,B)))
=> ~ ( r2_hidden(E,C)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k6_pencil_1(A,B)))
=> ~ ( r2_hidden(F,D)
& r1_pencil_1(k6_pencil_1(A,B),E,F) ) ) ) ) ) ) ) ) ) ).
fof(t20_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ( r1_pencil_3(A,B,C,D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(E,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(E,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [F] :
( m1_pencil_2(F,A,B)
=> ! [G] :
( m1_pencil_2(G,A,B)
=> ( ( F = k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),E,C)
& G = k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),E,D) )
=> r1_pencil_3(A,B,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t21_pencil_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ( r1_xboole_0(C,D)
=> ( r1_pencil_3(A,B,C,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] :
( m1_subset_1(G,A)
& G != k3_pencil_1(A,E)
& ! [H] :
( m1_subset_1(H,A)
=> ( H != G
=> k1_funct_1(E,H) = k1_funct_1(F,H) ) )
& ! [H] :
( m1_subset_1(H,u1_struct_0(k2_pencil_1(A,B,G)))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k2_pencil_1(A,B,G)))
=> ( ( k1_funct_1(E,G) = k1_struct_0(k2_pencil_1(A,B,G),H)
& k1_funct_1(F,G) = k1_struct_0(k2_pencil_1(A,B,G),I) )
=> r1_pencil_1(k2_pencil_1(A,B,G),H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v9_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C)))
=> ! [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)) )
=> ~ ( m1_pencil_2(k4_card_3(E),A,B)
& m1_pencil_2(k4_card_3(F),A,B)
& r6_pboole(A,E,k2_polynom1(A,F,C,k1_struct_0(k2_pencil_1(A,B,C),D)))
& ~ r2_hidden(D,k1_funct_1(F,C))
& ! [G] :
( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
=> ~ ( k1_funct_1(G,np__1) = k4_card_3(E)
& k1_funct_1(G,k3_finseq_1(G)) = k4_card_3(F)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k4_finseq_1(G))
=> m1_pencil_2(k1_funct_1(G,H),A,B) ) )
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,H)
=> ( r1_xreal_0(k3_finseq_1(G),H)
| ! [I] :
( m1_pencil_2(I,A,B)
=> ! [J] :
( m1_pencil_2(J,A,B)
=> ( ( I = k1_funct_1(G,H)
& J = k1_funct_1(G,k1_nat_1(H,np__1)) )
=> ( r1_xboole_0(I,J)
& r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v9_pencil_1(k2_pencil_1(A,B,C)) )
=> ! [C] :
( m1_pencil_2(C,A,B)
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ( r1_xboole_0(C,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] :
( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
& k1_funct_1(G,np__1) = C
& k1_funct_1(G,k3_finseq_1(G)) = D
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k4_finseq_1(G))
=> m1_pencil_2(k1_funct_1(G,H),A,B) ) )
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,H)
=> ( r1_xreal_0(k3_finseq_1(G),H)
| ! [I] :
( m1_pencil_2(I,A,B)
=> ! [J] :
( m1_pencil_2(J,A,B)
=> ( ( I = k1_funct_1(G,H)
& J = k1_funct_1(G,k1_nat_1(H,np__1)) )
=> ( r1_xboole_0(I,J)
& r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ! [E] :
( m1_pencil_2(E,A,B)
=> ! [F] :
( ( ~ v13_pencil_1(F)
& v16_pencil_1(F,A)
& m4_pboole(F,A,k12_pralg_1(A,B)) )
=> ! [G] :
( ( ~ v13_pencil_1(G)
& v16_pencil_1(G,A)
& m4_pboole(G,A,k12_pralg_1(A,B)) )
=> ! [H] :
( ( ~ v13_pencil_1(H)
& v16_pencil_1(H,A)
& m4_pboole(H,A,k12_pralg_1(A,B)) )
=> ! [I] :
( ( ~ v13_pencil_1(I)
& v16_pencil_1(I,A)
& m4_pboole(I,A,k12_pralg_1(A,B)) )
=> ( ( D = k4_card_3(F)
& E = k4_card_3(G)
& k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,D) = k4_card_3(H)
& k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,E) = k4_card_3(I)
& k3_pencil_1(A,F) = k3_pencil_1(A,G) )
=> k3_pencil_1(A,H) = k3_pencil_1(A,I) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t25_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,A,A)
& v3_funct_2(D,A,A)
& m2_relset_1(D,A,A)
& ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( k8_funct_2(A,A,D,E) = F
<=> ! [G] :
( m1_pencil_2(G,A,B)
=> ! [H] :
( ( ~ v13_pencil_1(H)
& v16_pencil_1(H,A)
& m4_pboole(H,A,k12_pralg_1(A,B)) )
=> ! [I] :
( ( ~ v13_pencil_1(I)
& v16_pencil_1(I,A)
& m4_pboole(I,A,k12_pralg_1(A,B)) )
=> ( ( G = k4_card_3(H)
& k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,G) = k4_card_3(I)
& k3_pencil_1(A,H) = E )
=> k3_pencil_1(A,I) = F ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,A)
& v3_funct_2(D,A,A)
& m2_relset_1(D,A,A) )
=> ( D = k2_pencil_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( k8_funct_2(A,A,D,E) = F
<=> ! [G] :
( m1_pencil_2(G,A,B)
=> ! [H] :
( ( ~ v13_pencil_1(H)
& v16_pencil_1(H,A)
& m4_pboole(H,A,k12_pralg_1(A,B)) )
=> ! [I] :
( ( ~ v13_pencil_1(I)
& v16_pencil_1(I,A)
& m4_pboole(I,A,k12_pralg_1(A,B)) )
=> ( ( G = k4_card_3(H)
& k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,G) = k4_card_3(I)
& k3_pencil_1(A,H) = E )
=> k3_pencil_1(A,I) = F ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( ( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B)) )
=> ( m1_pencil_2(k4_card_3(D),A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D)))))
& m2_relset_1(E,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))) )
=> ( E = k3_pencil_3(A,B,C,D)
<=> ( v1_pencil_2(E,k2_pencil_1(A,B,k3_pencil_1(A,D)),k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))
& ! [F] :
( m1_pboole(F,A)
=> ( ( m1_subset_1(F,u1_struct_0(k6_pencil_1(A,B)))
& r2_hidden(F,k4_card_3(D)) )
=> ! [G] :
( m1_pboole(G,A)
=> ( G = k1_funct_1(C,F)
=> k1_funct_1(G,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))) = k1_funct_1(E,k1_funct_1(F,k3_pencil_1(A,D))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t26_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( m1_pencil_2(D,A,B)
=> ! [E] :
( m1_pencil_2(E,A,B)
=> ( ( r1_xboole_0(D,E)
& r1_pencil_3(A,B,D,E) )
=> ! [F] :
( ( ~ v13_pencil_1(F)
& v16_pencil_1(F,A)
& m4_pboole(F,A,k12_pralg_1(A,B)) )
=> ! [G] :
( ( ~ v13_pencil_1(G)
& v16_pencil_1(G,A)
& m4_pboole(G,A,k12_pralg_1(A,B)) )
=> ( ( k4_card_3(F) = D
& k4_card_3(G) = E )
=> k3_pencil_3(A,B,C,F) = k3_pencil_3(A,B,C,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t27_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( ( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B)) )
=> ! [E] :
( ( ~ v13_pencil_1(E)
& v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,B)) )
=> ( ( m1_pencil_2(k4_card_3(D),A,B)
& m1_pencil_2(k4_card_3(E),A,B)
& k3_pencil_1(A,D) = k3_pencil_1(A,E) )
=> k3_pencil_3(A,B,C,D) = k3_pencil_3(A,B,C,E) ) ) ) ) ) ) ) ).
fof(d5_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D))))
& m2_relset_1(E,u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D)))) )
=> ( E = k4_pencil_3(A,B,C,D)
<=> ! [F] :
( ( ~ v13_pencil_1(F)
& v16_pencil_1(F,A)
& m4_pboole(F,A,k12_pralg_1(A,B)) )
=> ( ( m1_pencil_2(k4_card_3(F),A,B)
& k3_pencil_1(A,F) = D )
=> E = k3_pencil_3(A,B,C,F) ) ) ) ) ) ) ) ) ) ).
fof(t28_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,A,A)
& v3_funct_2(D,A,A)
& m2_relset_1(D,A,A)
& ? [E] :
( v1_funcop_1(E)
& m1_pboole(E,A)
& ! [F] :
( m1_subset_1(F,A)
=> ( v1_funct_1(k1_funct_1(E,F))
& v1_funct_2(k1_funct_1(E,F),u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
& m2_relset_1(k1_funct_1(E,F),u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
& m2_relset_1(G,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F)))) )
=> ( G = k1_funct_1(E,F)
=> v1_pencil_2(G,k2_pencil_1(A,B,F),k2_pencil_1(A,B,k8_funct_2(A,A,D,F))) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k6_pencil_1(A,B)))
=> ! [H] :
( m1_pboole(H,A)
=> ( H = G
=> ! [I] :
( m1_pboole(I,A)
=> ( I = k8_funct_2(u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)),C,G)
=> ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
& m2_relset_1(J,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F)))) )
=> ( J = k1_funct_1(E,F)
=> k1_funct_1(I,k8_funct_2(A,A,D,F)) = k1_funct_1(J,k1_funct_1(H,F)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_pencil_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_pboole(B,A)
& m1_pboole(C,A) )
=> m2_subset_1(k1_pencil_3(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k2_pencil_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v15_pencil_1(B)
& m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
=> ( v1_funct_1(k2_pencil_3(A,B,C))
& v1_funct_2(k2_pencil_3(A,B,C),A,A)
& v3_funct_2(k2_pencil_3(A,B,C),A,A)
& m2_relset_1(k2_pencil_3(A,B,C),A,A) ) ) ).
fof(dt_k3_pencil_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v15_pencil_1(B)
& m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B)) )
=> ( v1_funct_1(k3_pencil_3(A,B,C,D))
& v1_funct_2(k3_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D)))))
& m2_relset_1(k3_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))) ) ) ).
fof(dt_k4_pencil_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v15_pencil_1(B)
& m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
& m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
& m1_subset_1(D,A) )
=> ( v1_funct_1(k4_pencil_3(A,B,C,D))
& v1_funct_2(k4_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D))))
& m2_relset_1(k4_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D)))) ) ) ).
fof(d1_pencil_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> k1_pencil_3(A,B,C) = k1_card_1(a_3_0_pencil_3(A,B,C)) ) ) ) ).
fof(fraenkel_a_3_0_pencil_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_pboole(C,B)
& m1_pboole(D,B) )
=> ( r2_hidden(A,a_3_0_pencil_3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& k1_funct_1(C,E) != k1_funct_1(D,E) ) ) ) ).
%------------------------------------------------------------------------------