SET007 Axioms: SET007+878.ax
%------------------------------------------------------------------------------
% File : SET007+878 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Pencil Spaces, Grassmann Spaces, and Generalized Veronese 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 2 unt; 0 def)
% Number of atoms : 1223 ( 50 equ)
% Maximal formula atoms : 34 ( 18 avg)
% Number of connectives : 1356 ( 198 ~; 12 |; 929 &)
% ( 13 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 14 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 6 con; 0-5 aty)
% Number of variables : 224 ( 216 !; 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(fc1_pencil_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k8_pencil_4(A)) ) ).
fof(fc2_pencil_4,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ~ v1_xboole_0(k9_pencil_4(A,B)) ) ).
fof(fc3_pencil_4,axiom,
! [A,B] :
( ~ v1_realset1(B)
=> ( ~ v1_xboole_0(k9_pencil_4(A,B))
& ~ v1_realset1(k9_pencil_4(A,B)) ) ) ).
fof(fc4_pencil_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ~ v1_xboole_0(k10_pencil_4(A,B)) ) ).
fof(fc5_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k11_pencil_4(A))
& v1_pre_topc(k11_pencil_4(A)) ) ) ).
fof(fc6_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k11_pencil_4(A))
& v1_pre_topc(k11_pencil_4(A))
& ~ v3_pencil_1(k11_pencil_4(A)) ) ) ).
fof(fc7_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k11_pencil_4(A))
& v1_pre_topc(k11_pencil_4(A))
& ~ v3_pencil_1(k11_pencil_4(A))
& ~ v4_pencil_1(k11_pencil_4(A)) ) ) ).
fof(fc8_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& v5_pencil_1(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k11_pencil_4(A))
& v1_pre_topc(k11_pencil_4(A))
& ~ v3_pencil_1(k11_pencil_4(A))
& v5_pencil_1(k11_pencil_4(A)) ) ) ).
fof(fc9_pencil_4,axiom,
! [A] :
( ( v6_pencil_1(A)
& l1_pre_topc(A) )
=> ( v1_pre_topc(k11_pencil_4(A))
& v6_pencil_1(k11_pencil_4(A)) ) ) ).
fof(t1_pencil_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(B,A)
& r1_xreal_0(np__3,B)
& r1_xreal_0(B,k1_nat_1(A,np__1))
& ~ r1_xreal_0(np__2,A) ) ) ) ).
fof(t2_pencil_4,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,k4_card_1(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k4_card_1(C) != B ) ) ) ) ).
fof(t3_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> m1_vectsp_4(C,A,k2_vectsp_4(A,B)) ) ) ) ).
fof(t4_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,k2_vectsp_4(A,B))
=> m1_vectsp_4(C,A,B) ) ) ) ).
fof(t5_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> m1_vectsp_4(k2_vectsp_4(A,C),A,B) ) ) ) ).
fof(t6_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ( m1_vectsp_4(k2_vectsp_4(A,C),A,B)
=> m1_vectsp_4(C,A,B) ) ) ) ) ).
fof(d1_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_vectsp_5(A,B)))
=> ( ( m1_vectsp_4(C,A,D)
=> ( E = k1_pencil_4(A,B,C,D)
<=> ! [F] :
( ( v11_vectsp_1(F,A)
& m1_vectsp_4(F,A,B) )
=> ( r2_hidden(F,E)
<=> ( m1_vectsp_4(C,A,F)
& m1_vectsp_4(F,A,D) ) ) ) ) )
& ( ~ m1_vectsp_4(C,A,D)
=> ( E = k1_pencil_4(A,B,C,D)
<=> E = k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t7_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ( ( m1_vectsp_4(C,A,D)
& m1_vectsp_4(E,A,F)
& k2_vectsp_4(A,C) = k2_vectsp_4(A,E)
& k2_vectsp_4(A,D) = k2_vectsp_4(A,F) )
=> k1_pencil_4(A,B,C,D) = k1_pencil_4(A,B,E,F) ) ) ) ) ) ) ) ).
fof(d2_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> k2_pencil_4(A,B,C,D) = k4_xboole_0(k1_pencil_4(A,B,C,D),k2_tarski(k2_vectsp_4(A,C),k2_vectsp_4(A,D))) ) ) ) ) ).
fof(t8_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ( ( m1_vectsp_4(C,A,D)
& m1_vectsp_4(E,A,F)
& k2_vectsp_4(A,C) = k2_vectsp_4(A,E)
& k2_vectsp_4(A,D) = k2_vectsp_4(A,F) )
=> k2_pencil_4(A,B,C,D) = k2_pencil_4(A,B,E,F) ) ) ) ) ) ) ) ).
fof(d3_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k3_pencil_4(A,B,C,D,E) = k3_xboole_0(k2_pencil_4(A,B,C,D),k2_vectsp_9(A,B,E)) ) ) ) ) ) ).
fof(t9_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ( r2_hidden(F,k3_pencil_4(A,B,D,E,C))
=> ( m1_vectsp_4(D,A,F)
& m1_vectsp_4(F,A,E) ) ) ) ) ) ) ) ) ).
fof(t10_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ! [G] :
( m1_vectsp_4(G,A,B)
=> ( ( m1_vectsp_4(D,A,E)
& m1_vectsp_4(F,A,G)
& k2_vectsp_4(A,D) = k2_vectsp_4(A,F)
& k2_vectsp_4(A,E) = k2_vectsp_4(A,G) )
=> k3_pencil_4(A,B,D,E,C) = k3_pencil_4(A,B,F,G,C) ) ) ) ) ) ) ) ) ).
fof(d4_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_vectsp_9(A,B,C))))
=> ( D = k4_pencil_4(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( m1_vectsp_4(F,A,B)
& ? [G] :
( m1_vectsp_4(G,A,B)
& m1_vectsp_4(F,A,G)
& k1_nat_1(k1_vectsp_9(A,F),np__1) = C
& k1_vectsp_9(A,G) = k1_nat_1(C,np__1)
& E = k3_pencil_4(A,B,F,G,C) ) ) ) ) ) ) ) ) ).
fof(t11_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(k1_vectsp_9(A,B),C)
& v1_xboole_0(k4_pencil_4(A,B,C)) ) ) ) ) ).
fof(t12_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,G)
& k1_nat_1(k1_vectsp_9(A,C),np__1) = G
& k1_vectsp_9(A,D) = k1_nat_1(G,np__1)
& r2_hidden(E,k3_pencil_4(A,B,C,D,G))
& r2_hidden(F,k3_pencil_4(A,B,C,D,G)) )
=> ( r1_xreal_0(k1_vectsp_9(A,B),G)
| E = F
| ( k2_vectsp_5(A,B,E,F) = k2_vectsp_4(A,C)
& k1_vectsp_5(A,B,E,F) = k2_vectsp_4(A,D) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( C != k1_rlvect_1(B)
=> k1_vectsp_9(A,k1_vectsp_7(A,B,k1_struct_0(B,C))) = np__1 ) ) ) ) ).
fof(t14_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ~ r1_rlvect_1(C,D)
=> k1_vectsp_9(A,k1_vectsp_5(A,B,C,k1_vectsp_7(A,B,k1_struct_0(B,D)))) = k1_nat_1(k1_vectsp_9(A,C),np__1) ) ) ) ) ) ).
fof(t15_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( ( v1_vectsp_7(k2_struct_0(B,D,E),A,B)
& k2_vectsp_5(A,B,C,k1_vectsp_7(A,B,k2_struct_0(B,D,E))) = k1_vectsp_4(A,B) )
=> ( r1_rlvect_1(C,D)
| r1_rlvect_1(C,E)
| D = E
| k1_vectsp_9(A,k1_vectsp_5(A,B,C,k1_vectsp_7(A,B,k2_struct_0(B,D,E)))) = k1_nat_1(k1_vectsp_9(A,C),np__2) ) ) ) ) ) ) ) ).
fof(t16_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ( m1_vectsp_4(C,A,D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& k1_nat_1(k1_vectsp_9(A,C),np__1) = E
& k1_vectsp_9(A,D) = k1_nat_1(E,np__1) )
=> ( r1_xreal_0(k1_vectsp_9(A,B),E)
| ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( r1_rlvect_1(D,F)
=> ( r1_rlvect_1(C,F)
| r2_hidden(k1_vectsp_5(A,B,C,k1_vectsp_7(A,B,k1_struct_0(B,F))),k3_pencil_4(A,B,C,D,E)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ( m1_vectsp_4(C,A,D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& ~ r1_xreal_0(k1_vectsp_9(A,B),E)
& k1_nat_1(k1_vectsp_9(A,C),np__1) = E
& k1_vectsp_9(A,D) = k1_nat_1(E,np__1)
& v1_realset1(k3_pencil_4(A,B,C,D,E)) ) ) ) ) ) ) ) ).
fof(d5_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_pencil_4(A,B,C) = g1_pre_topc(k2_vectsp_9(A,B,C),k4_pencil_4(A,B,C)) ) ) ) ).
fof(t18_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,k1_vectsp_9(A,B))
& v3_struct_0(k5_pencil_4(A,B,C)) ) ) ) ) ).
fof(t19_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(k1_vectsp_9(A,B),C)
& v3_pencil_1(k5_pencil_4(A,B,C)) ) ) ) ) ).
fof(t20_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(k1_vectsp_9(A,B),C)
& r1_xreal_0(np__3,k1_vectsp_9(A,B))
& v4_pencil_1(k5_pencil_4(A,B,C)) ) ) ) ) ).
fof(t21_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k1_vectsp_9(A,B),C)
| v5_pencil_1(k5_pencil_4(A,B,C)) ) ) ) ) ) ).
fof(t22_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k1_vectsp_9(A,B),C)
| v6_pencil_1(k5_pencil_4(A,B,C)) ) ) ) ) ) ).
fof(t23_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(np__3,k1_vectsp_9(A,B)) )
=> ( r1_xreal_0(k1_vectsp_9(A,B),C)
| ( ~ v3_struct_0(k5_pencil_4(A,B,C))
& ~ v3_pencil_1(k5_pencil_4(A,B,C))
& ~ v4_pencil_1(k5_pencil_4(A,B,C))
& v5_pencil_1(k5_pencil_4(A,B,C))
& v6_pencil_1(k5_pencil_4(A,B,C))
& l1_pre_topc(k5_pencil_4(A,B,C)) ) ) ) ) ) ) ).
fof(d6_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(k2_vectsp_9(A,B,C))))
=> ( E = k6_pencil_4(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ? [G] :
( m1_vectsp_4(G,A,B)
& k1_vectsp_9(A,G) = D
& F = k2_vectsp_9(A,G,C) ) ) ) ) ) ) ) ) ).
fof(t24_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(D,k1_vectsp_9(A,B))
& v1_xboole_0(k6_pencil_4(A,B,C,D)) ) ) ) ) ) ).
fof(t25_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& v2_matrlin(C,A)
& l4_vectsp_1(C,A) )
=> ( k2_vectsp_4(A,B) = k2_vectsp_4(A,C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_vectsp_9(A,B,D) = k2_vectsp_9(A,C,D) ) ) ) ) ) ).
fof(t26_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k1_vectsp_9(A,B))
& r1_tarski(k2_vectsp_9(A,B,D),k2_vectsp_9(A,C,D)) )
=> k2_vectsp_4(A,B) = k2_vectsp_4(A,C) ) ) ) ) ) ).
fof(d7_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k7_pencil_4(A,B,C,D) = g1_pre_topc(k2_vectsp_9(A,B,C),k6_pencil_4(A,B,C,D)) ) ) ) ) ).
fof(t27_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,k1_vectsp_9(A,B))
& v3_struct_0(k7_pencil_4(A,B,C,D)) ) ) ) ) ) ).
fof(t28_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(D,k1_vectsp_9(A,B))
& v3_pencil_1(k7_pencil_4(A,B,C,D)) ) ) ) ) ) ).
fof(t29_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(k1_vectsp_9(A,B),D)
& v4_pencil_1(k7_pencil_4(A,B,C,D)) ) ) ) ) ) ).
fof(t30_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(D,k1_vectsp_9(A,B)) )
=> ( r1_xreal_0(D,C)
| v5_pencil_1(k7_pencil_4(A,B,C,D)) ) ) ) ) ) ) ).
fof(t31_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k1_vectsp_9(A,B)) )
=> v6_pencil_1(k7_pencil_4(A,B,C,k1_nat_1(C,np__1))) ) ) ) ) ).
fof(t32_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k1_vectsp_9(A,B),k1_nat_1(C,np__1))
| ( ~ v3_struct_0(k7_pencil_4(A,B,C,k1_nat_1(C,np__1)))
& ~ v3_pencil_1(k7_pencil_4(A,B,C,k1_nat_1(C,np__1)))
& ~ v4_pencil_1(k7_pencil_4(A,B,C,k1_nat_1(C,np__1)))
& v5_pencil_1(k7_pencil_4(A,B,C,k1_nat_1(C,np__1)))
& v6_pencil_1(k7_pencil_4(A,B,C,k1_nat_1(C,np__1)))
& l1_pre_topc(k7_pencil_4(A,B,C,k1_nat_1(C,np__1))) ) ) ) ) ) ) ).
fof(d8_pencil_4,axiom,
! [A,B] :
( B = k8_pencil_4(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D,E] :
( r2_hidden(D,A)
& r2_hidden(E,A)
& C = k2_tarski(D,E) ) ) ) ).
fof(d9_pencil_4,axiom,
! [A,B,C] :
( C = k9_pencil_4(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( r2_hidden(E,B)
& D = k2_tarski(A,E) ) ) ) ).
fof(d10_pencil_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k8_pencil_4(A))))
=> ( C = k10_pencil_4(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F] :
( m1_subset_1(F,k1_zfmisc_1(A))
& r2_hidden(E,A)
& r2_hidden(F,B)
& D = k9_pencil_4(E,F) ) ) ) ) ) ).
fof(d11_pencil_4,axiom,
! [A] :
( l1_pre_topc(A)
=> k11_pencil_4(A) = g1_pre_topc(k8_pencil_4(u1_struct_0(A)),k10_pencil_4(u1_struct_0(A),u1_pre_topc(A))) ) ).
fof(dt_k1_pencil_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_4(C,A,B)
& m1_vectsp_4(D,A,B) )
=> m1_subset_1(k1_pencil_4(A,B,C,D),k1_zfmisc_1(k3_vectsp_5(A,B))) ) ).
fof(dt_k2_pencil_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_4(C,A,B)
& m1_vectsp_4(D,A,B) )
=> m1_subset_1(k2_pencil_4(A,B,C,D),k1_zfmisc_1(k3_vectsp_5(A,B))) ) ).
fof(dt_k3_pencil_4,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_4(C,A,B)
& m1_vectsp_4(D,A,B)
& m1_subset_1(E,k5_numbers) )
=> m1_subset_1(k3_pencil_4(A,B,C,D,E),k1_zfmisc_1(k2_vectsp_9(A,B,E))) ) ).
fof(dt_k4_pencil_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k4_pencil_4(A,B,C),k1_zfmisc_1(k1_zfmisc_1(k2_vectsp_9(A,B,C)))) ) ).
fof(dt_k5_pencil_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( v1_pre_topc(k5_pencil_4(A,B,C))
& l1_pre_topc(k5_pencil_4(A,B,C)) ) ) ).
fof(dt_k6_pencil_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> m1_subset_1(k6_pencil_4(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(k2_vectsp_9(A,B,C)))) ) ).
fof(dt_k7_pencil_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> ( v1_pre_topc(k7_pencil_4(A,B,C,D))
& l1_pre_topc(k7_pencil_4(A,B,C,D)) ) ) ).
fof(dt_k8_pencil_4,axiom,
$true ).
fof(dt_k9_pencil_4,axiom,
$true ).
fof(dt_k10_pencil_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k10_pencil_4(A,B),k1_zfmisc_1(k1_zfmisc_1(k8_pencil_4(A)))) ) ).
fof(dt_k11_pencil_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_pre_topc(k11_pencil_4(A))
& l1_pre_topc(k11_pencil_4(A)) ) ) ).
fof(dt_k12_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& v5_pencil_1(A)
& v6_pencil_1(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k12_pencil_4(A))
& v1_pre_topc(k12_pencil_4(A))
& ~ v3_pencil_1(k12_pencil_4(A))
& ~ v4_pencil_1(k12_pencil_4(A))
& v5_pencil_1(k12_pencil_4(A))
& v6_pencil_1(k12_pencil_4(A))
& l1_pre_topc(k12_pencil_4(A)) ) ) ).
fof(redefinition_k12_pencil_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& v5_pencil_1(A)
& v6_pencil_1(A)
& l1_pre_topc(A) )
=> k12_pencil_4(A) = k11_pencil_4(A) ) ).
%------------------------------------------------------------------------------