SET007 Axioms: SET007+366.ax
%------------------------------------------------------------------------------
% File : SET007+366 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Monoid of Multisets and Subsets
% 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 : monoid_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 111 ( 5 unt; 0 def)
% Number of atoms : 701 ( 79 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 750 ( 160 ~; 0 |; 324 &)
% ( 8 <=>; 258 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 78 ( 78 usr; 8 con; 0-8 aty)
% Number of variables : 374 ( 371 !; 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_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ~ v3_struct_0(k9_monoid_1(A,B)) ) ).
fof(fc2_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k9_monoid_1(A,B))
& v1_monoid_0(k9_monoid_1(A,B)) ) ) ).
fof(fc3_monoid_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B))) )
=> ~ v1_xboole_0(k2_relat_1(C)) ) ).
fof(fc4_monoid_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_finset_1(B) )
=> ( v1_relat_1(k8_relat_1(A,B))
& v1_finset_1(k8_relat_1(A,B)) ) ) ).
fof(fc5_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ~ v3_struct_0(k22_monoid_1(A)) ) ).
fof(d1_monoid_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] : k1_monoid_1(A,B,C,D) = k5_funct_6(A,k4_tarski(B,C),D) ) ).
fof(t1_monoid_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D,E] :
( ( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
& B = k1_binop_1(A,C,D)
& r2_hidden(E,k1_relat_1(B)) )
=> k1_monoid_1(A,C,D,E) = k1_funct_1(B,E) ) ) ) ).
fof(t2_monoid_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] : k5_relat_1(B,k7_relat_1(A,C)) = k5_relat_1(k8_relat_1(C,B),A) ) ) ).
fof(d2_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E,F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C))
& m2_relset_1(F,k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C)) )
=> ( F = k8_monoid_1(A,B,C,D,E)
<=> ! [G] :
( m2_fraenkel(G,E,A,k1_fraenkel(E,A))
=> ! [H] :
( m2_fraenkel(H,E,B,k1_fraenkel(E,B))
=> k2_binop_1(k1_fraenkel(E,A),k1_fraenkel(E,B),k1_fraenkel(E,C),F,G,H) = k3_monoid_1(E,A,B,C,D,G,H) ) ) ) ) ) ) ) ) ).
fof(t3_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E,F] :
( ( v1_funct_1(F)
& v1_funct_2(F,E,A)
& m2_relset_1(F,E,A) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,E,B)
& m2_relset_1(G,E,B) )
=> ! [H] :
( r2_hidden(H,E)
=> k1_monoid_1(k8_monoid_1(A,B,C,D,E),F,G,H) = k1_binop_1(D,k1_funct_1(F,H),k1_funct_1(G,H)) ) ) ) ) ) ) ) ).
fof(t4_monoid_1,axiom,
! [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] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( v1_binop_1(C,B)
=> k3_monoid_1(A,B,B,B,C,D,E) = k3_monoid_1(A,B,B,B,C,E,D) ) ) ) ) ) ).
fof(t5_monoid_1,axiom,
! [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] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,B)
& m2_relset_1(F,A,B) )
=> ( v2_binop_1(C,B)
=> k3_monoid_1(A,B,B,B,C,k3_monoid_1(A,B,B,B,C,D,E),F) = k3_monoid_1(A,B,B,B,C,D,k3_monoid_1(A,B,B,B,C,E,F)) ) ) ) ) ) ) ).
fof(t6_monoid_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( r3_binop_1(B,C,D)
=> ( k6_monoid_1(A,B,B,B,D,C,E) = E
& k7_monoid_1(A,B,B,B,D,E,C) = E ) ) ) ) ) ) ).
fof(t7_monoid_1,axiom,
! [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] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( v3_binop_1(C,B)
=> k3_monoid_1(A,B,B,B,C,D,D) = D ) ) ) ) ).
fof(t8_monoid_1,axiom,
! [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) )
=> ( v1_binop_1(C,B)
=> v1_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t9_monoid_1,axiom,
! [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) )
=> ( v2_binop_1(C,B)
=> v2_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t10_monoid_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( r3_binop_1(B,C,D)
=> r3_binop_1(k1_fraenkel(A,B),k5_monoid_1(B,A,C),k8_monoid_1(B,B,B,D,A)) ) ) ) ) ).
fof(t11_monoid_1,axiom,
! [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) )
=> ( v1_setwiseo(C,B)
=> ( k3_binop_1(k1_fraenkel(A,B),k8_monoid_1(B,B,B,C,A)) = k5_monoid_1(B,A,k3_binop_1(B,C))
& v1_setwiseo(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ) ).
fof(t12_monoid_1,axiom,
! [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) )
=> ( v3_binop_1(C,B)
=> v3_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t13_monoid_1,axiom,
! [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) )
=> ( v5_monoid_0(C,B)
=> v5_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t14_monoid_1,axiom,
! [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) )
=> ( v8_monoid_0(C,B)
=> v8_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t15_monoid_1,axiom,
! [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) )
=> ( v9_monoid_0(C,B)
=> v9_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ).
fof(t16_monoid_1,axiom,
! [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] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( r1_lattice2(B,C,D)
=> r1_lattice2(k1_fraenkel(A,B),k8_monoid_1(B,B,B,C,A),k8_monoid_1(B,B,B,D,A)) ) ) ) ) ).
fof(t17_monoid_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( ~ v1_xboole_0(F)
=> ! [G] :
( ~ v1_xboole_0(G)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(B,C),D)
& m2_relset_1(H,k2_zfmisc_1(B,C),D) )
=> ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,k2_zfmisc_1(E,F),G)
& m2_relset_1(I,k2_zfmisc_1(E,F),G) )
=> ( r1_tarski(H,I)
=> r1_tarski(k8_monoid_1(B,C,D,H,A),k8_monoid_1(E,F,G,I,A)) ) ) ) ) ) ) ) ) ) ).
fof(d3_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( v2_group_1(A)
=> k9_monoid_1(A,B) = g1_vectsp_1(k1_fraenkel(B,u1_struct_0(A)),k8_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B),k5_monoid_1(u1_struct_0(A),B,k3_binop_1(u1_struct_0(A),u1_group_1(A)))) )
& ( ~ v2_group_1(A)
=> k9_monoid_1(A,B) = g1_group_1(k1_fraenkel(B,u1_struct_0(A)),k8_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B)) ) ) ) ).
fof(t18_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ( u1_struct_0(k9_monoid_1(B,A)) = k1_fraenkel(A,u1_struct_0(B))
& u1_group_1(k9_monoid_1(B,A)) = k8_monoid_1(u1_struct_0(B),u1_struct_0(B),u1_struct_0(B),u1_group_1(B),A) ) ) ).
fof(t19_monoid_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(C)
& l1_group_1(C) )
=> ( m1_subset_1(A,u1_struct_0(k9_monoid_1(C,B)))
<=> ( v1_funct_1(A)
& v1_funct_2(A,B,u1_struct_0(C))
& m2_relset_1(A,B,u1_struct_0(C)) ) ) ) ).
fof(t20_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
=> ( k1_relat_1(C) = A
& r1_tarski(k2_relat_1(C),u1_struct_0(B)) ) ) ) ).
fof(t21_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_monoid_1(B,A)))
=> ( ! [E] :
( r2_hidden(E,A)
=> k1_funct_1(C,E) = k1_funct_1(D,E) )
=> C = D ) ) ) ) ).
fof(t22_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_monoid_1(B,A)))
=> ! [E] :
( m1_subset_1(E,A)
=> k10_monoid_1(B,A,k1_group_1(k9_monoid_1(B,A),C,D),E) = k1_group_1(B,k10_monoid_1(B,A,C,E),k10_monoid_1(B,A,D,E)) ) ) ) ) ) ).
fof(t23_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& l1_group_1(B) )
=> u1_vectsp_1(k11_monoid_1(B,A)) = k5_monoid_1(u1_struct_0(B),A,k3_binop_1(u1_struct_0(B),u1_group_1(B))) ) ).
fof(t24_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( v7_group_1(A)
=> v7_group_1(k9_monoid_1(A,B)) )
& ( v4_group_1(A)
=> v4_group_1(k9_monoid_1(A,B)) )
& ( v10_monoid_0(A)
=> v10_monoid_0(k9_monoid_1(A,B)) )
& ( v13_monoid_0(A)
=> v13_monoid_0(k9_monoid_1(A,B)) )
& ( v16_monoid_0(A)
=> v16_monoid_0(k9_monoid_1(A,B)) )
& ( v17_monoid_0(A)
=> v17_monoid_0(k9_monoid_1(A,B)) ) ) ) ).
fof(t25_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_monoid_0(C,B) )
=> m2_monoid_0(k9_monoid_1(C,A),k9_monoid_1(B,A)) ) ) ).
fof(t26_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_monoid_0(C,B) )
=> ( r2_hidden(k3_binop_1(u1_struct_0(B),u1_group_1(B)),u1_struct_0(C))
=> m3_monoid_0(k9_monoid_1(C,A),k11_monoid_1(B,A)) ) ) ) ).
fof(d4_monoid_1,axiom,
! [A] : k13_monoid_1(A) = k12_monoid_1(k8_monoid_0,A) ).
fof(t27_monoid_1,axiom,
! [A] :
( u1_struct_0(k13_monoid_1(A)) = k1_fraenkel(A,k5_numbers)
& u1_group_1(k13_monoid_1(A)) = k8_monoid_1(k5_numbers,k5_numbers,k5_numbers,k47_binop_2,A)
& u1_vectsp_1(k13_monoid_1(A)) = k5_monoid_1(k5_numbers,A,np__0) ) ).
fof(t28_monoid_1,axiom,
! [A,B] :
( m1_subset_1(A,u1_struct_0(k13_monoid_1(B)))
<=> ( v1_funct_1(A)
& v1_funct_2(A,B,k5_numbers)
& m2_relset_1(A,B,k5_numbers) ) ) ).
fof(t29_monoid_1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
=> ( k1_relat_1(B) = A
& r1_tarski(k2_relat_1(B),k5_numbers) ) ) ).
fof(t30_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
=> ! [D] :
( m1_subset_1(D,A)
=> k15_monoid_1(A,k4_monoid_0(k13_monoid_1(A),B,C),D) = k1_nat_1(k15_monoid_1(A,B,D),k15_monoid_1(A,C,D)) ) ) ) ) ).
fof(t31_monoid_1,axiom,
! [A,B] : m1_subset_1(k5_funct_3(A,B),u1_struct_0(k13_monoid_1(B))) ).
fof(d5_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k18_monoid_1(A,B) = k16_monoid_1(k6_domain_1(A,B),A) ) ) ).
fof(t32_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( k15_monoid_1(A,k18_monoid_1(A,B),B) = np__1
& ( C != B
=> k15_monoid_1(A,k18_monoid_1(A,B),C) = np__0 ) ) ) ) ) ).
fof(t33_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
=> ( ! [D] :
( m1_subset_1(D,A)
=> k15_monoid_1(A,B,D) = k15_monoid_1(A,C,D) )
=> B = C ) ) ) ) ).
fof(d6_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v1_vectsp_1(B)
& m3_monoid_0(B,k13_monoid_1(A)) )
=> ( B = k19_monoid_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
=> ( r2_hidden(C,u1_struct_0(B))
<=> v1_finset_1(k10_relat_1(C,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,np__0)))) ) ) ) ) ).
fof(t34_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> m1_subset_1(k18_monoid_1(A,B),u1_struct_0(k19_monoid_1(A))) ) ) ).
fof(t35_monoid_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> k1_relat_1(k8_relat_1(k1_tarski(A),k7_finseq_1(C,k9_finseq_1(A)))) = k2_xboole_0(k1_relat_1(k8_relat_1(k1_tarski(A),C)),k6_domain_1(k5_numbers,k1_nat_1(k3_finseq_1(C),np__1))) ) ) ).
fof(t36_monoid_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,C)
=> ( A != B
=> k1_relat_1(k8_relat_1(k1_tarski(A),k7_finseq_1(D,k9_finseq_1(B)))) = k1_relat_1(k8_relat_1(k1_tarski(A),D)) ) ) ) ).
fof(d7_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
=> ( C = k20_monoid_1(A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> k15_monoid_1(A,C,D) = k4_card_1(k1_relat_1(k8_relat_1(k6_domain_1(A,D),B))) ) ) ) ) ) ).
fof(t37_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k15_monoid_1(A,k20_monoid_1(A,k6_finseq_1(A)),B) = np__0 ) ) ).
fof(t38_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k20_monoid_1(A,k6_finseq_1(A)) = k17_monoid_1(A,np__0) ) ).
fof(t39_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k20_monoid_1(A,k12_finseq_1(A,B)) = k18_monoid_1(A,B) ) ) ).
fof(t40_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k20_monoid_1(A,k1_monoid_0(A,C,k12_finseq_1(A,B))) = k4_monoid_0(k13_monoid_1(A),k20_monoid_1(A,C),k18_monoid_1(A,B)) ) ) ) ).
fof(t41_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k20_monoid_1(A,k1_monoid_0(A,B,C)) = k4_monoid_0(k13_monoid_1(A),k20_monoid_1(A,B),k20_monoid_1(A,C)) ) ) ) ).
fof(t42_monoid_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),C) = A
& ! [D] :
( m1_subset_1(D,B)
=> ( D != C
=> k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),D) = np__0 ) ) ) ) ) ) ).
fof(t43_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> m1_subset_1(k20_monoid_1(A,B),u1_struct_0(k19_monoid_1(A))) ) ) ).
fof(t44_monoid_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ~ ( m1_subset_1(A,u1_struct_0(k19_monoid_1(B)))
& ! [C] :
( m2_finseq_1(C,B)
=> A != k20_monoid_1(B,C) ) ) ) ).
fof(d8_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C))
& m2_relset_1(E,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C)) )
=> ( E = k21_monoid_1(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)))
=> k8_funct_2(k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C),E,F) = k2_funct_2(k2_zfmisc_1(A,B),C,D,k2_zfmisc_1(k2_domain_1(k1_zfmisc_1(A),k1_zfmisc_1(B),F),k3_domain_1(k1_zfmisc_1(A),k1_zfmisc_1(B),F))) ) ) ) ) ) ) ) ).
fof(t45_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F) = k2_funct_2(k2_zfmisc_1(A,B),C,D,k2_zfmisc_1(E,F)) ) ) ) ) ) ) ).
fof(t46_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> ! [G,H] :
( ( r2_hidden(G,E)
& r2_hidden(H,F) )
=> r2_hidden(k1_binop_1(D,G,H),k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F)) ) ) ) ) ) ) ) ).
fof(t48_monoid_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(C,C),C)
& m2_relset_1(D,k2_zfmisc_1(C,C),C) )
=> ( v1_binop_1(D,C)
=> k2_funct_2(k2_zfmisc_1(C,C),C,D,k2_zfmisc_1(A,B)) = k2_funct_2(k2_zfmisc_1(C,C),C,D,k2_zfmisc_1(B,A)) ) ) ) ).
fof(t49_monoid_1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(D,D),D)
& m2_relset_1(E,k2_zfmisc_1(D,D),D) )
=> ( v2_binop_1(E,D)
=> k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(A,B)),C)) = k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(A,k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(B,C)))) ) ) ) ).
fof(t50_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( v1_binop_1(B,A)
=> v1_binop_1(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) ).
fof(t51_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( v2_binop_1(B,A)
=> v2_binop_1(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) ).
fof(t52_monoid_1,axiom,
! [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)
=> ( r3_binop_1(B,D,C)
=> ( k2_funct_2(k2_zfmisc_1(B,B),B,C,k2_zfmisc_1(k6_domain_1(B,D),A)) = k3_xboole_0(B,A)
& k2_funct_2(k2_zfmisc_1(B,B),B,C,k2_zfmisc_1(A,k6_domain_1(B,D))) = k3_xboole_0(B,A) ) ) ) ) ) ).
fof(t53_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(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] :
( m1_subset_1(C,A)
=> ( r3_binop_1(A,C,B)
=> ( r3_binop_1(k1_zfmisc_1(A),k6_domain_1(A,C),k21_monoid_1(A,A,A,B))
& v1_setwiseo(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A))
& k3_binop_1(k1_zfmisc_1(A),k21_monoid_1(A,A,A,B)) = k6_domain_1(A,C) ) ) ) ) ) ).
fof(t54_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( v1_setwiseo(B,A)
=> ( v1_setwiseo(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A))
& r3_binop_1(k1_zfmisc_1(A),k6_domain_1(A,k3_binop_1(A,B)),k21_monoid_1(A,A,A,B))
& k3_binop_1(k1_zfmisc_1(A),k21_monoid_1(A,A,A,B)) = k6_domain_1(A,k3_binop_1(A,B)) ) ) ) ) ).
fof(t55_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( v9_monoid_0(B,A)
=> v9_monoid_0(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) ).
fof(d9_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( ( v2_group_1(A)
=> k22_monoid_1(A) = g1_vectsp_1(k1_zfmisc_1(u1_struct_0(A)),k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A)),k6_domain_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A)))) )
& ( ~ v2_group_1(A)
=> k22_monoid_1(A) = g1_group_1(k1_zfmisc_1(u1_struct_0(A)),k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A))) ) ) ) ).
fof(t56_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( u1_struct_0(k22_monoid_1(A)) = k1_zfmisc_1(u1_struct_0(A))
& u1_group_1(k22_monoid_1(A)) = k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A)) ) ) ).
fof(t57_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> u1_vectsp_1(k23_monoid_1(A)) = k6_domain_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A))) ) ).
fof(t58_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( ( v7_group_1(A)
=> v7_group_1(k22_monoid_1(A)) )
& ( v4_group_1(A)
=> v4_group_1(k22_monoid_1(A)) )
& ( v17_monoid_0(A)
=> v17_monoid_0(k22_monoid_1(A)) ) ) ) ).
fof(s1_monoid_1,axiom,
( ! [A] :
~ ( r2_hidden(A,f1_s1_monoid_1)
& ! [B] :
( m1_subset_1(B,f2_s1_monoid_1)
=> ~ p1_s1_monoid_1(A,B) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s1_monoid_1,f2_s1_monoid_1)
& m2_relset_1(A,f1_s1_monoid_1,f2_s1_monoid_1)
& ! [B] :
( r2_hidden(B,f1_s1_monoid_1)
=> p1_s1_monoid_1(B,k1_funct_1(A,B)) ) ) ) ).
fof(dt_k1_monoid_1,axiom,
$true ).
fof(dt_k2_monoid_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
& m1_relset_1(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
& m1_subset_1(F,B)
& m1_subset_1(G,C)
& m1_subset_1(H,A) )
=> m1_subset_1(k2_monoid_1(A,B,C,D,E,F,G,H),D) ) ).
fof(redefinition_k2_monoid_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
& m1_relset_1(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
& m1_subset_1(F,B)
& m1_subset_1(G,C)
& m1_subset_1(H,A) )
=> k2_monoid_1(A,B,C,D,E,F,G,H) = k1_monoid_1(E,F,G,H) ) ).
fof(dt_k3_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B)
& v1_funct_1(G)
& v1_funct_2(G,A,C)
& m1_relset_1(G,A,C) )
=> m2_fraenkel(k3_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) ).
fof(redefinition_k3_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B)
& v1_funct_1(G)
& v1_funct_2(G,A,C)
& m1_relset_1(G,A,C) )
=> k3_monoid_1(A,B,C,D,E,F,G) = k3_funcop_1(E,F,G) ) ).
fof(dt_k4_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> m2_finseq_1(k4_monoid_1(A,B,C),A) ) ).
fof(redefinition_k4_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> k4_monoid_1(A,B,C) = k2_finseq_2(B,C) ) ).
fof(dt_k5_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,A) )
=> m2_fraenkel(k5_monoid_1(A,B,C),B,A,k1_fraenkel(B,A)) ) ).
fof(redefinition_k5_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,A) )
=> k5_monoid_1(A,B,C) = k2_funcop_1(B,C) ) ).
fof(dt_k6_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& m1_subset_1(F,B)
& v1_funct_1(G)
& v1_funct_2(G,A,C)
& m1_relset_1(G,A,C) )
=> m2_fraenkel(k6_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) ).
fof(redefinition_k6_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& m1_subset_1(F,B)
& v1_funct_1(G)
& v1_funct_2(G,A,C)
& m1_relset_1(G,A,C) )
=> k6_monoid_1(A,B,C,D,E,F,G) = k5_funcop_1(E,F,G) ) ).
fof(dt_k7_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B)
& m1_subset_1(G,C) )
=> m2_fraenkel(k7_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) ).
fof(redefinition_k7_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m1_relset_1(E,k2_zfmisc_1(B,C),D)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B)
& m1_subset_1(G,C) )
=> k7_monoid_1(A,B,C,D,E,F,G) = k4_funcop_1(E,F,G) ) ).
fof(dt_k8_monoid_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k8_monoid_1(A,B,C,D,E))
& v1_funct_2(k8_monoid_1(A,B,C,D,E),k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C))
& m2_relset_1(k8_monoid_1(A,B,C,D,E),k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C)) ) ) ).
fof(dt_k9_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> l1_group_1(k9_monoid_1(A,B)) ) ).
fof(dt_k10_monoid_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B)))
& m1_subset_1(D,B) )
=> m1_subset_1(k10_monoid_1(A,B,C,D),u1_struct_0(A)) ) ).
fof(redefinition_k10_monoid_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B)))
& m1_subset_1(D,B) )
=> k10_monoid_1(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k11_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k11_monoid_1(A,B))
& v1_vectsp_1(k11_monoid_1(A,B))
& v1_vectsp_2(k11_monoid_1(A,B))
& v1_monoid_0(k11_monoid_1(A,B))
& l1_vectsp_1(k11_monoid_1(A,B)) ) ) ).
fof(redefinition_k11_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> k11_monoid_1(A,B) = k9_monoid_1(A,B) ) ).
fof(dt_k12_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v16_monoid_0(A)
& v17_monoid_0(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k12_monoid_1(A,B))
& v4_group_1(k12_monoid_1(A,B))
& v7_group_1(k12_monoid_1(A,B))
& v1_vectsp_1(k12_monoid_1(A,B))
& v1_vectsp_2(k12_monoid_1(A,B))
& v1_monoid_0(k12_monoid_1(A,B))
& v16_monoid_0(k12_monoid_1(A,B))
& v17_monoid_0(k12_monoid_1(A,B))
& l1_vectsp_1(k12_monoid_1(A,B)) ) ) ).
fof(redefinition_k12_monoid_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v16_monoid_0(A)
& v17_monoid_0(A)
& l1_group_1(A) )
=> k12_monoid_1(A,B) = k9_monoid_1(A,B) ) ).
fof(dt_k13_monoid_1,axiom,
! [A] :
( ~ v3_struct_0(k13_monoid_1(A))
& v4_group_1(k13_monoid_1(A))
& v7_group_1(k13_monoid_1(A))
& v1_vectsp_1(k13_monoid_1(A))
& v1_vectsp_2(k13_monoid_1(A))
& v1_monoid_0(k13_monoid_1(A))
& v16_monoid_0(k13_monoid_1(A))
& v17_monoid_0(k13_monoid_1(A))
& l1_vectsp_1(k13_monoid_1(A)) ) ).
fof(dt_k14_monoid_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) )
=> ( ~ v1_xboole_0(k14_monoid_1(A,B))
& m1_subset_1(k14_monoid_1(A,B),k1_zfmisc_1(k5_numbers)) ) ) ).
fof(redefinition_k14_monoid_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) )
=> k14_monoid_1(A,B) = k2_relat_1(B) ) ).
fof(dt_k15_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
& m1_subset_1(C,A) )
=> m2_subset_1(k15_monoid_1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(redefinition_k15_monoid_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
& m1_subset_1(C,A) )
=> k15_monoid_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k16_monoid_1,axiom,
! [A,B] : m1_subset_1(k16_monoid_1(A,B),u1_struct_0(k13_monoid_1(B))) ).
fof(redefinition_k16_monoid_1,axiom,
! [A,B] : k16_monoid_1(A,B) = k4_funct_3(A,B) ).
fof(dt_k17_monoid_1,axiom,
! [A,B] :
( m1_subset_1(B,k5_numbers)
=> m1_subset_1(k17_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) ).
fof(redefinition_k17_monoid_1,axiom,
! [A,B] :
( m1_subset_1(B,k5_numbers)
=> k17_monoid_1(A,B) = k2_funcop_1(A,B) ) ).
fof(dt_k18_monoid_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k18_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) ).
fof(dt_k19_monoid_1,axiom,
! [A] :
( ~ v3_struct_0(k19_monoid_1(A))
& v1_vectsp_1(k19_monoid_1(A))
& m3_monoid_0(k19_monoid_1(A),k13_monoid_1(A)) ) ).
fof(dt_k20_monoid_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> m1_subset_1(k20_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) ).
fof(dt_k21_monoid_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k21_monoid_1(A,B,C,D))
& v1_funct_2(k21_monoid_1(A,B,C,D),k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C))
& m2_relset_1(k21_monoid_1(A,B,C,D),k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C)) ) ) ).
fof(dt_k22_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> l1_group_1(k22_monoid_1(A)) ) ).
fof(dt_k23_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k23_monoid_1(A))
& v1_vectsp_1(k23_monoid_1(A))
& v1_vectsp_2(k23_monoid_1(A))
& l1_vectsp_1(k23_monoid_1(A)) ) ) ).
fof(redefinition_k23_monoid_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> k23_monoid_1(A) = k22_monoid_1(A) ) ).
fof(t47_monoid_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F) = a_6_0_monoid_1(A,B,C,D,E,F) ) ) ) ) ) ) ).
fof(fraenkel_a_6_0_monoid_1,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m2_relset_1(E,k2_zfmisc_1(B,C),D)
& m1_subset_1(F,k1_zfmisc_1(B))
& m1_subset_1(G,k1_zfmisc_1(C)) )
=> ( r2_hidden(A,a_6_0_monoid_1(B,C,D,E,F,G))
<=> ? [H,I] :
( m1_subset_1(H,B)
& m1_subset_1(I,C)
& A = k2_binop_1(B,C,D,E,H,I)
& r2_hidden(H,F)
& r2_hidden(I,G) ) ) ) ).
%------------------------------------------------------------------------------