SET007 Axioms: SET007+443.ax
%------------------------------------------------------------------------------
% File : SET007+443 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Monoid of Endomorphisms of Universal and Many Sorted Algebras
% 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 : endalg [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 5 unt; 0 def)
% Number of atoms : 367 ( 28 equ)
% Maximal formula atoms : 15 ( 8 avg)
% Number of connectives : 387 ( 65 ~; 0 |; 216 &)
% ( 13 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 30 ( 30 usr; 1 con; 0-6 aty)
% Number of variables : 98 ( 95 !; 3 ?)
% 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_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( ~ v3_struct_0(k3_endalg(A))
& v1_vectsp_1(k3_endalg(A)) ) ) ).
fof(fc2_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( ~ v3_struct_0(k3_endalg(A))
& v2_group_1(k3_endalg(A))
& v4_group_1(k3_endalg(A))
& v1_vectsp_1(k3_endalg(A))
& v6_vectsp_1(k3_endalg(A))
& v8_vectsp_1(k3_endalg(A)) ) ) ).
fof(fc3_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( ~ v3_struct_0(k6_endalg(A,B))
& v1_vectsp_1(k6_endalg(A,B)) ) ) ).
fof(fc4_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( ~ v3_struct_0(k6_endalg(A,B))
& v2_group_1(k6_endalg(A,B))
& v4_group_1(k6_endalg(A,B))
& v1_vectsp_1(k6_endalg(A,B))
& v6_vectsp_1(k6_endalg(A,B))
& v8_vectsp_1(k6_endalg(A,B)) ) ) ).
fof(rc1_endalg,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v8_vectsp_1(A) ) ).
fof(rc2_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A)
& ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> ? [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v1_relat_1(C)
& v1_funct_1(C)
& ~ v1_xboole_0(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_partfun1(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& v1_endalg(C,A,B) ) ) ).
fof(d1_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
=> ( B = k1_endalg(A)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_hidden(C,B)
<=> r1_alg_1(A,A,C) ) ) ) ) ) ).
fof(t1_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r1_tarski(k1_endalg(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) ).
fof(t2_endalg,axiom,
$true ).
fof(t3_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r2_hidden(k6_partfun1(u1_struct_0(A)),k1_endalg(A)) ) ).
fof(t4_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> r2_hidden(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),k1_endalg(A)) ) ) ) ).
fof(d2_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A))
& m2_relset_1(B,k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A)) )
=> ( B = k2_endalg(A)
<=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> k2_binop_1(k1_endalg(A),k1_endalg(A),k1_endalg(A),B,C,D) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,D) ) ) ) ) ) ).
fof(d3_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_vectsp_1(B)
& l1_vectsp_1(B) )
=> ( B = k3_endalg(A)
<=> ( u1_struct_0(B) = k1_endalg(A)
& u1_group_1(B) = k2_endalg(A)
& u1_vectsp_1(B) = k6_partfun1(u1_struct_0(A)) ) ) ) ) ).
fof(t5_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_endalg(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_endalg(A)))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> ! [E] :
( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> ( ( B = D
& C = E )
=> k1_group_1(k3_endalg(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E) ) ) ) ) ) ) ).
fof(t6_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> k6_partfun1(u1_struct_0(A)) = k2_group_1(k3_endalg(A)) ) ).
fof(d4_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
=> ( C = k4_endalg(A,B)
<=> ( ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),C)
=> m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
& ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
=> ( r2_hidden(D,C)
<=> r1_msualg_3(A,B,B,D) ) ) ) ) ) ) ) ).
fof(t7_endalg,axiom,
$true ).
fof(t8_endalg,axiom,
$true ).
fof(t9_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> r1_tarski(k4_endalg(A,B),k4_card_3(k12_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)))) ) ) ).
fof(t10_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> r2_hidden(k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)),k4_endalg(A,B)) ) ) ).
fof(t11_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> r2_hidden(k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,C),k4_endalg(A,B)) ) ) ) ) ).
fof(t12_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m3_pboole(B,u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> ( B = k10_pboole(k1_tarski(np__0),C)
=> r2_hidden(B,k4_endalg(k7_msualg_1(A),k10_msualg_1(A))) ) ) ) ) ).
fof(d5_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B))
& m2_relset_1(C,k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B)) )
=> ( C = k5_endalg(A,B)
<=> ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> ! [E] :
( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> k2_binop_1(k4_endalg(A,B),k4_endalg(A,B),k4_endalg(A,B),C,D,E) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,E) ) ) ) ) ) ) ).
fof(d6_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_vectsp_1(C)
& l1_vectsp_1(C) )
=> ( C = k6_endalg(A,B)
<=> ( u1_struct_0(C) = k4_endalg(A,B)
& u1_group_1(C) = k5_endalg(A,B)
& u1_vectsp_1(C) = k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ) ) ).
fof(t13_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_endalg(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k6_endalg(A,B)))
=> ! [E] :
( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> ! [F] :
( m2_autalg_1(F,u1_struct_0(A),u4_msualg_1(A,B),k4_endalg(A,B))
=> ( ( C = E
& D = F )
=> k1_group_1(k6_endalg(A,B),C,D) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),E,F) ) ) ) ) ) ) ) ).
fof(t14_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)) = k2_group_1(k6_endalg(A,B)) ) ) ).
fof(t15_endalg,axiom,
$true ).
fof(t16_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_endalg(A))
=> m3_pboole(k10_pboole(k1_tarski(np__0),B),u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ) ).
fof(d7_endalg,axiom,
$true ).
fof(d8_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_endalg(C,A,B)
<=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_group_1(A)) = k2_group_1(B) ) ) ) ) ).
fof(d9_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_endalg(A,B,C)
<=> v2_funct_1(C) ) ) ) ) ).
fof(d10_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r2_endalg(A,B,C)
<=> k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ).
fof(d11_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r3_endalg(A,B,C)
<=> ( r2_endalg(A,B,C)
& r1_endalg(A,B,C) ) ) ) ) ) ).
fof(t17_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ( v1_funct_1(k6_partfun1(u1_struct_0(A)))
& v1_funct_2(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A))
& v1_group_6(k6_partfun1(u1_struct_0(A)),A,A)
& v1_endalg(k6_partfun1(u1_struct_0(A)),A,A)
& m2_relset_1(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(d12_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> ( r4_endalg(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& v1_endalg(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& r3_endalg(A,B,C) ) ) ) ) ).
fof(t18_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(B) = k1_endalg(A)
& ! [C] :
( r2_hidden(C,k1_endalg(A))
=> k1_funct_1(B,C) = k10_pboole(k1_tarski(np__0),C) ) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A))))
& v1_group_6(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
& v1_endalg(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
& m2_relset_1(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))) ) ) ) ) ).
fof(t19_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A))))
& v1_group_6(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
& v1_endalg(B,k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))
& m2_relset_1(B,u1_struct_0(k3_endalg(A)),u1_struct_0(k6_endalg(k7_msualg_1(A),k10_msualg_1(A)))) )
=> ( ! [C] :
( r2_hidden(C,k1_endalg(A))
=> k1_funct_1(B,C) = k10_pboole(k1_tarski(np__0),C) )
=> r3_endalg(k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A)),B) ) ) ) ).
fof(t20_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r4_endalg(k3_endalg(A),k6_endalg(k7_msualg_1(A),k10_msualg_1(A))) ) ).
fof(reflexivity_r4_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A)
& ~ v3_struct_0(B)
& v8_vectsp_1(B)
& l1_vectsp_1(B) )
=> r4_endalg(A,A) ) ).
fof(dt_k1_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> m1_fraenkel(k1_endalg(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k2_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( v1_funct_1(k2_endalg(A))
& v1_funct_2(k2_endalg(A),k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A))
& m2_relset_1(k2_endalg(A),k2_zfmisc_1(k1_endalg(A),k1_endalg(A)),k1_endalg(A)) ) ) ).
fof(dt_k3_endalg,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( v1_vectsp_1(k3_endalg(A))
& l1_vectsp_1(k3_endalg(A)) ) ) ).
fof(dt_k4_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> m1_autalg_1(k4_endalg(A,B),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ).
fof(dt_k5_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v1_funct_1(k5_endalg(A,B))
& v1_funct_2(k5_endalg(A,B),k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B))
& m2_relset_1(k5_endalg(A,B),k2_zfmisc_1(k4_endalg(A,B),k4_endalg(A,B)),k4_endalg(A,B)) ) ) ).
fof(dt_k6_endalg,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v1_vectsp_1(k6_endalg(A,B))
& l1_vectsp_1(k6_endalg(A,B)) ) ) ).
%------------------------------------------------------------------------------