SET007 Axioms: SET007+225.ax
%------------------------------------------------------------------------------
% File : SET007+225 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Fields
% 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 : realset2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 0 unt; 0 def)
% Number of atoms : 368 ( 71 equ)
% Maximal formula atoms : 35 ( 6 avg)
% Number of connectives : 328 ( 17 ~; 0 |; 154 &)
% ( 15 <=>; 142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 8 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 0 con; 1-6 aty)
% Number of variables : 162 ( 140 !; 22 ?)
% 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_realset2,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_rlvect_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v1_realset2(A)
& v2_realset2(A) ) ).
fof(rc2_realset2,axiom,
? [A] :
( l1_struct_0(A)
& v3_realset2(A) ) ).
fof(rc3_realset2,axiom,
? [A] :
( l3_vectsp_1(A)
& v3_vectsp_1(A)
& ~ v3_realset2(A) ) ).
fof(rc4_realset2,axiom,
? [A] :
( l3_vectsp_1(A)
& v3_vectsp_1(A)
& v4_realset2(A) ) ).
fof(d1_realset2,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( v1_realset2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_binop_1(u1_rlvect_1(A),B,u2_struct_0(A)) = B
& k1_binop_1(u1_rlvect_1(A),u2_struct_0(A),B) = B ) ) ) ) ).
fof(d2_realset2,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( v2_realset2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k1_binop_1(u1_rlvect_1(A),B,C) = u2_struct_0(A)
& k1_binop_1(u1_rlvect_1(A),C,B) = u2_struct_0(A) ) ) ) ) ).
fof(d3_realset2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v4_rlvect_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))
=> k1_binop_1(u1_rlvect_1(A),k1_binop_1(u1_rlvect_1(A),B,C),D) = k1_binop_1(u1_rlvect_1(A),B,k1_binop_1(u1_rlvect_1(A),C,D)) ) ) ) ) ) ).
fof(d4_realset2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v3_rlvect_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_binop_1(u1_rlvect_1(A),B,C) = k1_binop_1(u1_rlvect_1(A),C,B) ) ) ) ) ).
fof(d5_realset2,axiom,
! [A] :
( l1_struct_0(A)
=> ( v3_realset2(A)
<=> v1_realset1(u1_struct_0(A)) ) ) ).
fof(d6_realset2,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k4_xboole_0(A,k5_realset1(A,D)))
=> ! [F] :
( ( v3_vectsp_1(F)
& ~ v3_realset2(F)
& l3_vectsp_1(F) )
=> ( F = k1_realset2(A,B,C,D,E)
<=> ( A = u1_struct_0(F)
& B = u1_rlvect_1(F)
& C = u1_group_1(F)
& D = u2_struct_0(F)
& E = u1_vectsp_1(F) ) ) ) ) ) ) ) ) ).
fof(d7_realset2,axiom,
! [A] :
( l1_struct_0(A)
=> ( v3_realset2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> B = C ) ) ) ) ).
fof(d8_realset2,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( v4_realset2(A)
<=> ? [B] :
( ~ v1_realset1(B)
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B)
& ? [D] :
( m1_subset_1(D,B)
& ? [E] :
( m3_realset1(E,B,D)
& ? [F] :
( m1_subset_1(F,k4_xboole_0(B,k5_realset1(B,D)))
& A = k1_realset2(B,C,E,D,F)
& ~ v3_struct_0(g1_rlvect_1(B,C,D))
& v3_rlvect_1(g1_rlvect_1(B,C,D))
& v4_rlvect_1(g1_rlvect_1(B,C,D))
& v1_realset2(g1_rlvect_1(B,C,D))
& v2_realset2(g1_rlvect_1(B,C,D))
& l1_rlvect_1(g1_rlvect_1(B,C,D))
& ! [G] :
( ~ v1_xboole_0(G)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(G,G),G)
& m2_relset_1(H,k2_zfmisc_1(G,G),G) )
=> ! [I] :
( m1_subset_1(I,G)
=> ( ( G = k4_xboole_0(B,k5_realset1(B,D))
& I = F
& H = k4_realset1(B,D,E) )
=> ( ~ v3_struct_0(g1_rlvect_1(G,H,I))
& v3_rlvect_1(g1_rlvect_1(G,H,I))
& v4_rlvect_1(g1_rlvect_1(G,H,I))
& v1_realset2(g1_rlvect_1(G,H,I))
& v2_realset2(g1_rlvect_1(G,H,I))
& l1_rlvect_1(g1_rlvect_1(G,H,I)) ) ) ) ) )
& ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,B)
=> ( k1_funct_1(E,k4_tarski(G,k1_funct_1(C,k4_tarski(H,I)))) = k1_funct_1(C,k4_tarski(k1_funct_1(E,k4_tarski(G,H)),k1_funct_1(E,k4_tarski(G,I))))
& k1_funct_1(E,k4_tarski(k1_funct_1(C,k4_tarski(G,H)),I)) = k1_funct_1(C,k4_tarski(k1_funct_1(E,k4_tarski(G,I)),k1_funct_1(E,k4_tarski(H,I)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ~ v1_realset1(B)
=> ( B = k2_realset2(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B)
& ? [D] :
( m1_subset_1(D,B)
& ? [E] :
( m3_realset1(E,B,D)
& ? [F] :
( m1_subset_1(F,k4_xboole_0(B,k5_realset1(B,D)))
& A = k1_realset2(B,C,E,D,F) ) ) ) ) ) ) ) ).
fof(d10_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A))
& m2_relset_1(B,k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A)) )
=> ( B = k3_realset2(A)
<=> ? [C] :
( m1_subset_1(C,k2_realset2(A))
& ? [D] :
( m3_realset1(D,k2_realset2(A),C)
& ? [E] :
( m1_subset_1(E,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),C)))
& A = k1_realset2(k2_realset2(A),B,D,C,E) ) ) ) ) ) ) ).
fof(d11_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ( B = k4_realset2(A)
<=> ? [C] :
( m3_realset1(C,k2_realset2(A),B)
& ? [D] :
( m1_subset_1(D,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),B)))
& A = k1_realset2(k2_realset2(A),k3_realset2(A),C,B,D) ) ) ) ) ) ).
fof(d12_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m3_realset1(B,k2_realset2(A),k4_realset2(A))
=> ( B = k5_realset2(A)
<=> ? [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
& A = k1_realset2(k2_realset2(A),k3_realset2(A),B,k4_realset2(A),C) ) ) ) ) ).
fof(d13_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( B = k6_realset2(A)
<=> A = k1_realset2(k2_realset2(A),k3_realset2(A),k5_realset2(A),k4_realset2(A),B) ) ) ) ).
fof(t1_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A)))
& v3_rlvect_1(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A)))
& v4_rlvect_1(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A)))
& v1_realset2(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A)))
& v2_realset2(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A)))
& l1_rlvect_1(g1_rlvect_1(k2_realset2(A),k3_realset2(A),k4_realset2(A))) ) ) ).
fof(t2_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ! [D] :
( m1_subset_1(D,B)
=> ( ( B = k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))
& D = k6_realset2(A)
& C = k4_realset1(k2_realset2(A),k4_realset2(A),k5_realset2(A)) )
=> ( ~ v3_struct_0(g1_rlvect_1(B,C,D))
& v3_rlvect_1(g1_rlvect_1(B,C,D))
& v4_rlvect_1(g1_rlvect_1(B,C,D))
& v1_realset2(g1_rlvect_1(B,C,D))
& v2_realset2(g1_rlvect_1(B,C,D))
& l1_rlvect_1(g1_rlvect_1(B,C,D)) ) ) ) ) ) ) ).
fof(t3_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,k1_funct_1(k3_realset2(A),k4_tarski(C,D)))) = k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k1_funct_1(k5_realset2(A),k4_tarski(B,D))))
& k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k3_realset2(A),k4_tarski(B,C)),D)) = k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,D)),k1_funct_1(k5_realset2(A),k4_tarski(C,D)))) ) ) ) ) ) ).
fof(t4_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C),D) = k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),C,D)) ) ) ) ) ).
fof(t5_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C) = k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),C,B) ) ) ) ).
fof(t6_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ( k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,k4_realset2(A)) = B
& k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),k4_realset2(A),B) = B ) ) ) ).
fof(t7_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ? [C] :
( m1_subset_1(C,k2_realset2(A))
& k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C) = k4_realset2(A)
& k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),C,B) = k4_realset2(A) ) ) ) ).
fof(t8_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [D] :
( m1_subset_1(D,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),D)) = k1_funct_1(k5_realset2(A),k4_tarski(B,k1_funct_1(k5_realset2(A),k4_tarski(C,D)))) ) ) ) ) ).
fof(t9_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(k5_realset2(A),k4_tarski(B,C)) = k1_funct_1(k5_realset2(A),k4_tarski(C,B)) ) ) ) ).
fof(t10_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,k6_realset2(A))) = B
& k1_funct_1(k5_realset2(A),k4_tarski(k6_realset2(A),B)) = B ) ) ) ).
fof(t11_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ? [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
& k1_funct_1(k5_realset2(A),k4_tarski(B,C)) = k6_realset2(A)
& k1_funct_1(k5_realset2(A),k4_tarski(C,B)) = k6_realset2(A) ) ) ) ).
fof(d14_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_realset2(A),k2_realset2(A))
& m2_relset_1(B,k2_realset2(A),k2_realset2(A)) )
=> ( B = k7_realset2(A)
<=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> k1_funct_1(k3_realset2(A),k4_tarski(C,k8_funct_2(k2_realset2(A),k2_realset2(A),B,C))) = k4_realset2(A) ) ) ) ) ).
fof(t12_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ( k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C) = k4_realset2(A)
=> C = k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B) ) ) ) ) ).
fof(t13_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> B = k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B)) ) ) ).
fof(t14_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ( m1_subset_1(k1_funct_1(k3_realset2(A),k4_tarski(B,C)),k2_realset2(A))
& m1_subset_1(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k2_realset2(A))
& m1_subset_1(k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B),k2_realset2(A)) ) ) ) ) ).
fof(t15_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(B,k1_funct_1(k3_realset2(A),k4_tarski(C,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),D))))) = k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k1_funct_1(k7_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(B,D))))) ) ) ) ) ).
fof(t16_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k3_realset2(A),k4_tarski(B,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),C))),D)) = k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,D)),k1_funct_1(k7_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(C,D))))) ) ) ) ) ).
fof(t17_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(B,k4_realset2(A))) = k4_realset2(A) ) ) ).
fof(t18_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(k4_realset2(A),B)) = k4_realset2(A) ) ) ).
fof(t19_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> k1_funct_1(k7_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(B,C))) = k1_funct_1(k5_realset2(A),k4_tarski(B,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),C))) ) ) ) ).
fof(t20_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> k1_funct_1(k5_realset2(A),k4_tarski(k6_realset2(A),k4_realset2(A))) = k4_realset2(A) ) ).
fof(t21_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> k1_funct_1(k5_realset2(A),k4_tarski(k4_realset2(A),k6_realset2(A))) = k4_realset2(A) ) ).
fof(t22_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> m1_subset_1(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k2_realset2(A)) ) ) ) ).
fof(t23_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),D)) = k1_funct_1(k5_realset2(A),k4_tarski(B,k1_funct_1(k5_realset2(A),k4_tarski(C,D)))) ) ) ) ) ).
fof(t24_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> k1_funct_1(k5_realset2(A),k4_tarski(B,C)) = k1_funct_1(k5_realset2(A),k4_tarski(C,B)) ) ) ) ).
fof(t25_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,k6_realset2(A))) = B
& k1_funct_1(k5_realset2(A),k4_tarski(k6_realset2(A),B)) = B ) ) ) ).
fof(d15_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
& m2_relset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))) )
=> ( B = k8_realset2(A)
<=> ! [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(k5_realset2(A),k4_tarski(C,k8_funct_2(k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),B,C))) = k6_realset2(A) ) ) ) ) ).
fof(t26_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,C)) = k6_realset2(A)
=> C = k8_funct_2(k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k8_realset2(A),B) ) ) ) ) ).
fof(t27_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> B = k8_funct_2(k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k8_realset2(A),k8_funct_2(k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k8_realset2(A),B)) ) ) ).
fof(t28_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [C] :
( m1_subset_1(C,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( m1_subset_1(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
& m1_subset_1(k8_funct_2(k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k8_realset2(A),B),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))) ) ) ) ) ).
fof(t29_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> ( k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C) = k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,D)
=> C = D ) ) ) ) ) ).
fof(t30_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,C)) = k1_funct_1(k5_realset2(A),k4_tarski(B,D))
=> C = D ) ) ) ) ) ).
fof(dt_k1_realset2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_realset1(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& m1_subset_1(D,A)
& m1_subset_1(E,k4_xboole_0(A,k5_realset1(A,D))) )
=> ( v3_vectsp_1(k1_realset2(A,B,C,D,E))
& ~ v3_realset2(k1_realset2(A,B,C,D,E))
& l3_vectsp_1(k1_realset2(A,B,C,D,E)) ) ) ).
fof(dt_k2_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ~ v1_realset1(k2_realset2(A)) ) ).
fof(dt_k3_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k3_realset2(A))
& v1_funct_2(k3_realset2(A),k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A))
& m2_relset_1(k3_realset2(A),k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A)) ) ) ).
fof(dt_k4_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> m1_subset_1(k4_realset2(A),k2_realset2(A)) ) ).
fof(dt_k5_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> m3_realset1(k5_realset2(A),k2_realset2(A),k4_realset2(A)) ) ).
fof(dt_k6_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> m1_subset_1(k6_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))) ) ).
fof(dt_k7_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k7_realset2(A))
& v1_funct_2(k7_realset2(A),k2_realset2(A),k2_realset2(A))
& m2_relset_1(k7_realset2(A),k2_realset2(A),k2_realset2(A)) ) ) ).
fof(dt_k8_realset2,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k8_realset2(A))
& v1_funct_2(k8_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
& m2_relset_1(k8_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))) ) ) ).
%------------------------------------------------------------------------------