SET007 Axioms: SET007+262.ax
%------------------------------------------------------------------------------
% File : SET007+262 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Several Properties of Fields. Field Theory
% 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 : realset3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 6 unt; 0 def)
% Number of atoms : 242 ( 50 equ)
% Maximal formula atoms : 9 ( 5 avg)
% Number of connectives : 195 ( 0 ~; 1 |; 50 &)
% ( 11 <=>; 133 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 7 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 0 con; 1-6 aty)
% Number of variables : 133 ( 133 !; 0 ?)
% 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_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),k4_realset2(A)) = k4_realset2(A) ) ).
fof(t2_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(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),k6_realset2(A)) = k6_realset2(A) ) ).
fof(t3_realset3,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(k3_realset2(A),k4_tarski(B,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),C)))) = k1_funct_1(k3_realset2(A),k4_tarski(C,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B))) ) ) ) ).
fof(t4_realset3,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(k8_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(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),C)))) = 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))),k8_realset2(A),B))) ) ) ) ).
fof(t5_realset3,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(k3_realset2(A),k4_tarski(B,C))) = k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B),k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),C)) ) ) ) ).
fof(t6_realset3,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(k8_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(B,C))) = k1_funct_1(k5_realset2(A),k4_tarski(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),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),C))) ) ) ) ).
fof(t7_realset3,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))
=> ! [E] :
( m1_subset_1(E,k2_realset2(A))
=> ( k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),C)) = k1_funct_1(k3_realset2(A),k4_tarski(D,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),E)))
<=> k1_funct_1(k3_realset2(A),k4_tarski(B,E)) = k1_funct_1(k3_realset2(A),k4_tarski(C,D)) ) ) ) ) ) ) ).
fof(t8_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [E] :
( m1_subset_1(E,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(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),D))) = 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))),k8_realset2(A),E)))
<=> k1_funct_1(k5_realset2(A),k4_tarski(B,E)) = k1_funct_1(k5_realset2(A),k4_tarski(D,C)) ) ) ) ) ) ) ).
fof(t9_realset3,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)) = k4_realset2(A)
<=> ( B = k4_realset2(A)
| C = k4_realset2(A) ) ) ) ) ) ).
fof(t10_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [E] :
( m1_subset_1(E,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,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),D))),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))),k8_realset2(A),E))))) = k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k1_funct_1(k8_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(D,E))))) ) ) ) ) ) ).
fof(t11_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [E] :
( m1_subset_1(E,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(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),D))),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))),k8_realset2(A),E))))) = k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,E)),k1_funct_1(k5_realset2(A),k4_tarski(C,D)))),k1_funct_1(k8_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(D,E))))) ) ) ) ) ) ).
fof(d1_realset3,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 = k1_realset3(A)
<=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k2_realset2(A))
=> k1_funct_1(B,k4_tarski(C,D)) = k1_funct_1(k3_realset2(A),k4_tarski(C,k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),D))) ) ) ) ) ) ).
fof(t12_realset3,axiom,
$true ).
fof(t13_realset3,axiom,
$true ).
fof(t14_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k1_realset3(A),k4_tarski(B,B)) = k4_realset2(A) ) ) ).
fof(t15_realset3,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(k1_realset3(A),k4_tarski(C,D)))) = k1_funct_1(k1_realset3(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),k1_funct_1(k5_realset2(A),k4_tarski(B,D)))) ) ) ) ) ).
fof(t16_realset3,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(k1_realset3(A),k4_tarski(B,C)),k2_realset2(A)) ) ) ) ).
fof(t17_realset3,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(k1_realset3(A),k4_tarski(B,C)),D)) = k1_funct_1(k1_realset3(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,D)),k1_funct_1(k5_realset2(A),k4_tarski(C,D)))) ) ) ) ) ).
fof(t18_realset3,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(k1_realset3(A),k4_tarski(B,C)) = k1_funct_1(k7_realset2(A),k1_funct_1(k1_realset3(A),k4_tarski(C,B))) ) ) ) ).
fof(t19_realset3,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(k1_realset3(A),k4_tarski(k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B),C)) = k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),k2_binop_1(k2_realset2(A),k2_realset2(A),k2_realset2(A),k3_realset2(A),B,C)) ) ) ) ).
fof(t20_realset3,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))
=> ! [E] :
( m1_subset_1(E,k2_realset2(A))
=> ( k1_funct_1(k1_realset3(A),k4_tarski(B,C)) = k1_funct_1(k1_realset3(A),k4_tarski(D,E))
<=> k1_funct_1(k3_realset2(A),k4_tarski(B,E)) = k1_funct_1(k3_realset2(A),k4_tarski(C,D)) ) ) ) ) ) ) ).
fof(t21_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k1_realset3(A),k4_tarski(k4_realset2(A),B)) = k8_funct_2(k2_realset2(A),k2_realset2(A),k7_realset2(A),B) ) ) ).
fof(t22_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k1_realset3(A),k4_tarski(B,k4_realset2(A))) = B ) ) ).
fof(t23_realset3,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(k3_realset2(A),k4_tarski(B,C)) = D
<=> k1_funct_1(k1_realset3(A),k4_tarski(D,B)) = C ) ) ) ) ) ).
fof(t24_realset3,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(k3_realset2(A),k4_tarski(B,C)) = D
<=> k1_funct_1(k1_realset3(A),k4_tarski(D,C)) = B ) ) ) ) ) ).
fof(t25_realset3,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(k1_realset3(A),k4_tarski(B,k1_funct_1(k1_realset3(A),k4_tarski(C,D)))) = k1_funct_1(k3_realset2(A),k4_tarski(k1_funct_1(k1_realset3(A),k4_tarski(B,C)),D)) ) ) ) ) ).
fof(t26_realset3,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(k1_realset3(A),k4_tarski(B,k1_funct_1(k3_realset2(A),k4_tarski(C,D)))) = k1_funct_1(k1_realset3(A),k4_tarski(k1_funct_1(k1_realset3(A),k4_tarski(B,C)),D)) ) ) ) ) ).
fof(d2_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k2_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))),k2_realset2(A))
& m2_relset_1(B,k2_zfmisc_1(k2_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))),k2_realset2(A)) )
=> ( B = k2_realset3(A)
<=> ! [C] :
( m1_subset_1(C,k2_realset2(A))
=> ! [D] :
( m1_subset_1(D,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(B,k4_tarski(C,D)) = 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))),k8_realset2(A),D))) ) ) ) ) ) ).
fof(t27_realset3,axiom,
$true ).
fof(t28_realset3,axiom,
$true ).
fof(t29_realset3,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(k2_realset3(A),k4_tarski(B,B)) = k6_realset2(A) ) ) ).
fof(t30_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_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(k2_realset3(A),k4_tarski(B,C)),k2_realset2(A)) ) ) ) ).
fof(t31_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> k1_funct_1(k5_realset2(A),k4_tarski(B,k1_funct_1(k2_realset3(A),k4_tarski(C,D)))) = k1_funct_1(k2_realset3(A),k4_tarski(k1_funct_1(k5_realset2(A),k4_tarski(B,C)),D)) ) ) ) ) ).
fof(t32_realset3,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,k1_funct_1(k2_realset3(A),k4_tarski(k6_realset2(A),B)))) = k6_realset2(A)
& k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k2_realset3(A),k4_tarski(k6_realset2(A),B)),B)) = k6_realset2(A) ) ) ) ).
fof(t33_realset3,axiom,
$true ).
fof(t34_realset3,axiom,
$true ).
fof(t35_realset3,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(k2_realset3(A),k4_tarski(B,C)) = k1_funct_1(k8_realset2(A),k1_funct_1(k2_realset3(A),k4_tarski(C,B))) ) ) ) ).
fof(t36_realset3,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(k2_realset3(A),k4_tarski(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),C)) = k1_funct_1(k8_realset2(A),k1_funct_1(k5_realset2(A),k4_tarski(B,C))) ) ) ) ).
fof(t37_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ! [E] :
( m1_subset_1(E,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( k1_funct_1(k2_realset3(A),k4_tarski(B,D)) = k1_funct_1(k2_realset3(A),k4_tarski(C,E))
<=> k1_funct_1(k5_realset2(A),k4_tarski(B,E)) = k1_funct_1(k5_realset2(A),k4_tarski(D,C)) ) ) ) ) ) ) ).
fof(t38_realset3,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(k2_realset3(A),k4_tarski(k6_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),B) ) ) ).
fof(t39_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_realset2(A))
=> k1_funct_1(k2_realset3(A),k4_tarski(B,k6_realset2(A))) = B ) ) ).
fof(t40_realset3,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)) = D
<=> k1_funct_1(k2_realset3(A),k4_tarski(D,B)) = C ) ) ) ) ) ).
fof(t41_realset3,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,k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A))))
=> ( k1_funct_1(k5_realset2(A),k4_tarski(B,D)) = C
<=> k1_funct_1(k2_realset3(A),k4_tarski(C,D)) = B ) ) ) ) ) ).
fof(t42_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_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(k2_realset3(A),k4_tarski(B,k1_funct_1(k2_realset3(A),k4_tarski(C,D)))) = k1_funct_1(k5_realset2(A),k4_tarski(k1_funct_1(k2_realset3(A),k4_tarski(B,C)),D)) ) ) ) ) ).
fof(t43_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_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(k2_realset3(A),k4_tarski(B,k1_funct_1(k5_realset2(A),k4_tarski(C,D)))) = k1_funct_1(k2_realset3(A),k4_tarski(k1_funct_1(k2_realset3(A),k4_tarski(B,C)),D)) ) ) ) ) ).
fof(dt_k1_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k1_realset3(A))
& v1_funct_2(k1_realset3(A),k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A))
& m2_relset_1(k1_realset3(A),k2_zfmisc_1(k2_realset2(A),k2_realset2(A)),k2_realset2(A)) ) ) ).
fof(dt_k2_realset3,axiom,
! [A] :
( ( v4_realset2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k2_realset3(A))
& v1_funct_2(k2_realset3(A),k2_zfmisc_1(k2_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))),k2_realset2(A))
& m2_relset_1(k2_realset3(A),k2_zfmisc_1(k2_realset2(A),k4_xboole_0(k2_realset2(A),k5_realset1(k2_realset2(A),k4_realset2(A)))),k2_realset2(A)) ) ) ).
%------------------------------------------------------------------------------