SET007 Axioms: SET007+365.ax
%------------------------------------------------------------------------------
% File : SET007+365 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Monoids
% 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_0 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 227 ( 22 unt; 0 def)
% Number of atoms : 1278 ( 148 equ)
% Maximal formula atoms : 21 ( 5 avg)
% Number of connectives : 1270 ( 219 ~; 3 |; 573 &)
% ( 60 <=>; 415 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 65 ( 63 usr; 1 prp; 0-3 aty)
% Number of functors : 65 ( 65 usr; 25 con; 0-6 aty)
% Number of variables : 442 ( 409 !; 33 ?)
% 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_monoid_0,axiom,
? [A] :
( l1_struct_0(A)
& v1_monoid_0(A) ) ).
fof(rc2_monoid_0,axiom,
? [A] :
( l1_struct_0(A)
& v2_monoid_0(A) ) ).
fof(cc1_monoid_0,axiom,
! [A] :
( ( v1_monoid_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(cc2_monoid_0,axiom,
! [A] :
( l1_struct_0(A)
=> ( v2_monoid_0(A)
=> v1_monoid_0(A) ) ) ).
fof(cc3_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( v2_monoid_0(A)
=> v1_monoid_0(A) ) ) ).
fof(cc4_monoid_0,axiom,
! [A] :
( ( v2_monoid_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ) ).
fof(rc3_monoid_0,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v1_monoid_0(A)
& v2_monoid_0(A)
& v10_monoid_0(A)
& v13_monoid_0(A)
& v16_monoid_0(A)
& v17_monoid_0(A) ) ).
fof(cc5_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A) )
=> ( ~ v3_struct_0(A)
& v13_monoid_0(A) ) ) ) ).
fof(cc6_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v13_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A) ) ) ) ).
fof(cc7_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v13_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v11_monoid_0(A)
& v12_monoid_0(A) ) ) ) ).
fof(cc8_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v11_monoid_0(A)
& v12_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v13_monoid_0(A) ) ) ) ).
fof(cc9_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v16_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v14_monoid_0(A)
& v15_monoid_0(A) ) ) ) ).
fof(cc10_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v14_monoid_0(A)
& v15_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v16_monoid_0(A) ) ) ) ).
fof(cc11_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v13_monoid_0(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v16_monoid_0(A) ) ) ) ).
fof(cc12_monoid_0,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v1_vectsp_2(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A) ) ) ) ).
fof(fc1_monoid_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(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)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(g1_vectsp_1(A,B,C))
& v1_vectsp_1(g1_vectsp_1(A,B,C)) ) ) ).
fof(rc4_monoid_0,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v1_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v1_vectsp_2(A)
& v1_monoid_0(A)
& v2_monoid_0(A)
& v10_monoid_0(A)
& v11_monoid_0(A)
& v12_monoid_0(A)
& v13_monoid_0(A)
& v14_monoid_0(A)
& v15_monoid_0(A)
& v16_monoid_0(A)
& v17_monoid_0(A) ) ).
fof(cc13_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ~ v3_struct_0(B) ) ) ).
fof(rc5_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] :
( m1_monoid_0(B,A)
& v1_vectsp_1(B) ) ) ).
fof(cc14_monoid_0,axiom,
! [A] :
( ( v1_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> v1_monoid_0(B) ) ) ).
fof(cc15_monoid_0,axiom,
! [A] :
( ( v2_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( v1_monoid_0(B)
& v2_monoid_0(B) ) ) ) ).
fof(cc16_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v2_group_1(B)
& v6_vectsp_1(B)
& v8_vectsp_1(B) ) ) ) ).
fof(cc17_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v4_group_1(B) ) ) ) ).
fof(cc18_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v7_group_1(B) ) ) ) ).
fof(cc19_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v13_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v11_monoid_0(B)
& v12_monoid_0(B)
& v13_monoid_0(B) ) ) ) ).
fof(cc20_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v16_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B) ) ) ) ).
fof(cc21_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v17_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
& v17_monoid_0(B) ) ) ) ).
fof(rc6_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ? [B] :
( m1_monoid_0(B,A)
& ~ v3_struct_0(B)
& v2_group_1(B)
& v1_vectsp_1(B)
& v6_vectsp_1(B)
& v8_vectsp_1(B)
& v1_vectsp_2(B) ) ) ).
fof(rc7_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] :
( m2_monoid_0(B,A)
& v1_group_1(B) ) ) ).
fof(rc8_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ? [B] :
( m2_monoid_0(B,A)
& ~ v3_struct_0(B)
& v1_group_1(B) ) ) ).
fof(rc9_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ? [B] :
( m2_monoid_0(B,A)
& ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v10_monoid_0(B)
& v11_monoid_0(B)
& v12_monoid_0(B)
& v13_monoid_0(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B)
& v17_monoid_0(B) ) ) ).
fof(rc10_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] :
( m3_monoid_0(B,A)
& v1_vectsp_1(B) ) ) ).
fof(rc11_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ? [B] :
( m3_monoid_0(B,A)
& ~ v3_struct_0(B)
& v1_vectsp_1(B) ) ) ).
fof(rc12_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_2(A)
& l1_vectsp_1(A) )
=> ? [B] :
( m3_monoid_0(B,A)
& ~ v3_struct_0(B)
& v2_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v1_vectsp_1(B)
& v6_vectsp_1(B)
& v8_vectsp_1(B)
& v1_vectsp_2(B)
& v10_monoid_0(B)
& v11_monoid_0(B)
& v12_monoid_0(B)
& v13_monoid_0(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B)
& v17_monoid_0(B) ) ) ).
fof(cc22_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v1_monoid_0(B) ) ) ) ) ).
fof(cc23_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v1_monoid_0(B) ) ) ) ) ).
fof(cc24_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v1_monoid_0(B)
& v2_monoid_0(B) ) ) ) ) ).
fof(cc25_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v1_monoid_0(B)
& v2_monoid_0(B) ) ) ) ) ).
fof(cc26_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_2(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v2_group_1(B)
& v6_vectsp_1(B)
& v8_vectsp_1(B)
& v1_vectsp_2(B) ) ) ) ) ).
fof(cc27_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v7_group_1(B) ) ) ) ) ).
fof(cc28_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v7_group_1(B) ) ) ) ) ).
fof(cc29_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v4_group_1(B) ) ) ) ) ).
fof(cc30_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v4_group_1(B) ) ) ) ) ).
fof(cc31_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v10_monoid_0(B) ) ) ) ) ).
fof(cc32_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v10_monoid_0(B) ) ) ) ) ).
fof(cc33_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v16_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B) ) ) ) ) ).
fof(cc34_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v16_monoid_0(A)
& l1_group_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B) ) ) ) ) ).
fof(cc35_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_2(A)
& v17_monoid_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m3_monoid_0(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v2_group_1(B)
& v6_vectsp_1(B)
& v8_vectsp_1(B)
& v1_vectsp_2(B)
& v17_monoid_0(B) ) ) ) ) ).
fof(cc36_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_monoid_0(B,A)
=> ( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v13_monoid_0(B) )
=> ( ~ v3_struct_0(B)
& v2_group_1(B)
& v3_group_1(B)
& v14_monoid_0(B)
& v15_monoid_0(B)
& v16_monoid_0(B) ) ) ) ) ).
fof(d1_monoid_0,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_monoid_0(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ) ).
fof(d2_monoid_0,axiom,
! [A] :
( l1_struct_0(A)
=> ( v2_monoid_0(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) ) ) ) ).
fof(d3_monoid_0,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) )
=> ( v3_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( m1_subset_1(E,A)
& k2_binop_1(A,A,A,B,E,C) = D ) ) ) ) ) ) ).
fof(d4_monoid_0,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) )
=> ( v4_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( m1_subset_1(E,A)
& k2_binop_1(A,A,A,B,C,E) = D ) ) ) ) ) ) ).
fof(d5_monoid_0,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) )
=> ( v5_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( m1_subset_1(E,A)
& ? [F] :
( m1_subset_1(F,A)
& k2_binop_1(A,A,A,B,C,E) = D
& k2_binop_1(A,A,A,B,F,C) = D ) ) ) ) ) ) ) ).
fof(d6_monoid_0,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) )
=> ( v6_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
=> D = E ) ) ) ) ) ) ) ).
fof(d7_monoid_0,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) )
=> ( v7_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C)
=> D = E ) ) ) ) ) ) ) ).
fof(d8_monoid_0,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) )
=> ( v8_monoid_0(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
| k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C) )
=> D = E ) ) ) ) ) ) ) ).
fof(d9_monoid_0,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)
<=> ( v1_setwiseo(B,A)
& ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( k2_binop_1(A,A,A,B,C,D) = k3_binop_1(A,B)
=> ( C = D
& D = k3_binop_1(A,B) ) ) ) ) ) ) ) ) ).
fof(t1_monoid_0,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) )
=> ( v5_monoid_0(B,A)
<=> ( v3_monoid_0(B,A)
& v4_monoid_0(B,A) ) ) ) ) ).
fof(t2_monoid_0,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) )
=> ( v8_monoid_0(B,A)
<=> ( v6_monoid_0(B,A)
& v7_monoid_0(B,A) ) ) ) ) ).
fof(t3_monoid_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A))
& m2_relset_1(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A)) )
=> ( B = k2_funcop_1(k1_tarski(k4_tarski(A,A)),A)
& v1_setwiseo(B,k1_tarski(A))
& v1_binop_1(B,k1_tarski(A))
& v2_binop_1(B,k1_tarski(A))
& v3_binop_1(B,k1_tarski(A))
& v5_monoid_0(B,k1_tarski(A))
& v8_monoid_0(B,k1_tarski(A))
& v9_monoid_0(B,k1_tarski(A)) ) ) ).
fof(d10_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
<=> v1_setwiseo(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d11_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v7_group_1(A)
<=> v1_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d12_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v4_group_1(A)
<=> v2_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d13_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v10_monoid_0(A)
<=> v3_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d14_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v11_monoid_0(A)
<=> v3_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d15_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v12_monoid_0(A)
<=> v4_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d16_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v13_monoid_0(A)
<=> v5_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d17_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v14_monoid_0(A)
<=> v6_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d18_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v15_monoid_0(A)
<=> v7_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d19_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v16_monoid_0(A)
<=> v8_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(d20_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v17_monoid_0(A)
<=> v9_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ).
fof(t4_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
=> r3_binop_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_group_1(A)) ) ) ).
fof(t5_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_group_1(A,k3_binop_1(u1_struct_0(A),u1_group_1(A)),B) = B
& k1_group_1(A,B,k3_binop_1(u1_struct_0(A),u1_group_1(A))) = B ) ) ) ) ).
fof(t6_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v2_group_1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = C
& k1_group_1(A,C,B) = C ) ) ) ) ) ).
fof(t7_monoid_0,axiom,
$true ).
fof(t8_monoid_0,axiom,
$true ).
fof(t9_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v10_monoid_0(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,B) = B ) ) ) ).
fof(t10_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v11_monoid_0(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_group_1(A,D,B) = C ) ) ) ) ) ).
fof(t11_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v12_monoid_0(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_group_1(A,B,D) = C ) ) ) ) ) ).
fof(t12_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v13_monoid_0(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))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_group_1(A,B,D) = C
& k1_group_1(A,E,B) = C ) ) ) ) ) ) ).
fof(t13_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v14_monoid_0(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_group_1(A,B,C) = k1_group_1(A,B,D)
=> C = D ) ) ) ) ) ) ).
fof(t14_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v15_monoid_0(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_group_1(A,C,B) = k1_group_1(A,D,B)
=> C = D ) ) ) ) ) ) ).
fof(t15_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v16_monoid_0(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_group_1(A,B,C) = k1_group_1(A,B,D)
| k1_group_1(A,C,B) = k1_group_1(A,D,B) )
=> C = D ) ) ) ) ) ) ).
fof(t16_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v17_monoid_0(A)
<=> ( v1_setwiseo(u1_group_1(A),u1_struct_0(A))
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k3_binop_1(u1_struct_0(A),u1_group_1(A))
=> ( B = C
& C = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) ) ) ) ) ) ).
fof(t17_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v4_group_1(A)
=> ( v13_monoid_0(A)
<=> ( v2_group_1(A)
& v1_finseqop(u1_group_1(A),u1_struct_0(A)) ) ) ) ) ).
fof(d21_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v1_vectsp_2(A)
<=> r3_binop_1(u1_struct_0(A),u1_vectsp_1(A),u1_group_1(A)) ) ) ).
fof(t18_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v1_vectsp_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_group_1(A,u1_vectsp_1(A),B) = B
& k1_group_1(A,B,u1_vectsp_1(A)) = B ) ) ) ) ).
fof(t19_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v1_vectsp_2(A)
=> u1_vectsp_1(A) = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) ).
fof(d22_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( l1_vectsp_1(B)
=> ( m1_monoid_0(B,A)
<=> g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ) ).
fof(t20_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( u1_struct_0(B) = u1_struct_0(A)
& u1_group_1(B) = u1_group_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( C = E
& D = F )
=> k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t21_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_monoid_0(B,A)
=> ( ( v2_group_1(A)
=> v2_group_1(B) )
& ( v7_group_1(A)
=> v7_group_1(B) )
& ( v4_group_1(A)
=> v4_group_1(B) )
& ( v13_monoid_0(A)
=> v13_monoid_0(B) )
& ( v17_monoid_0(A)
=> v17_monoid_0(B) )
& ( v16_monoid_0(A)
=> v16_monoid_0(B) ) ) ) ) ).
fof(t22_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_vectsp_1(B)
& v1_vectsp_2(B)
& m1_monoid_0(B,A) )
=> ! [C] :
( ( v1_vectsp_1(C)
& v1_vectsp_2(C)
& m1_monoid_0(C,A) )
=> B = C ) ) ) ).
fof(d23_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( l1_group_1(B)
=> ( m2_monoid_0(B,A)
<=> r1_tarski(u1_group_1(B),u1_group_1(A)) ) ) ) ).
fof(d24_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( l1_vectsp_1(B)
=> ( m3_monoid_0(B,A)
<=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
& ! [C] :
( l1_vectsp_1(C)
=> ( A = C
=> u1_vectsp_1(B) = u1_vectsp_1(C) ) ) ) ) ) ) ).
fof(d25_monoid_0,axiom,
! [A] :
( l1_vectsp_1(A)
=> ! [B] :
( l1_vectsp_1(B)
=> ( m3_monoid_0(B,A)
<=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
& u1_vectsp_1(B) = u1_vectsp_1(A) ) ) ) ) ).
fof(t23_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( m3_monoid_0(B,A)
=> m2_monoid_0(B,A) ) ) ).
fof(t24_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_1(B) )
=> ( m2_monoid_0(A,A)
& m3_monoid_0(B,B) ) ) ) ).
fof(t25_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m3_monoid_0(C,A) )
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& r1_tarski(u1_struct_0(C),u1_struct_0(A)) ) ) ) ) ).
fof(t26_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) ).
fof(t27_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( C = E
& D = F )
=> k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ).
fof(t28_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_monoid_0(C,A) )
=> ( u1_struct_0(B) = u1_struct_0(C)
=> g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(C),u1_group_1(C)) ) ) ) ) ).
fof(t29_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m3_monoid_0(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m3_monoid_0(C,A) )
=> ( u1_struct_0(B) = u1_struct_0(C)
=> g1_vectsp_1(u1_struct_0(B),u1_group_1(B),u1_vectsp_1(B)) = g1_vectsp_1(u1_struct_0(C),u1_group_1(C),u1_vectsp_1(C)) ) ) ) ) ).
fof(t30_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_monoid_0(C,A) )
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
=> m5_monoid_0(B,A,C) ) ) ) ) ).
fof(t31_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m3_monoid_0(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m3_monoid_0(C,A) )
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
=> m7_monoid_0(B,A,C) ) ) ) ) ).
fof(t32_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( ( v2_group_1(A)
& r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B)) )
=> ( v2_group_1(B)
& k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k3_binop_1(u1_struct_0(B),u1_group_1(B)) ) ) ) ) ).
fof(t33_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_2(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m3_monoid_0(B,A) )
=> v1_vectsp_2(B) ) ) ).
fof(t34_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( v7_group_1(A)
=> v7_group_1(B) ) ) ) ).
fof(t35_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( v4_group_1(A)
=> v4_group_1(B) ) ) ) ).
fof(t36_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( v10_monoid_0(A)
=> v10_monoid_0(B) ) ) ) ).
fof(t37_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( v16_monoid_0(A)
=> v16_monoid_0(B) ) ) ) ).
fof(t38_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,A) )
=> ( ( r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B))
& v17_monoid_0(A) )
=> v17_monoid_0(B) ) ) ) ).
fof(t39_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_2(A)
& v17_monoid_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m3_monoid_0(B,A) )
=> v17_monoid_0(B) ) ) ).
fof(d26_monoid_0,axiom,
k5_monoid_0 = g1_group_1(k1_numbers,k33_binop_2) ).
fof(t40_monoid_0,axiom,
( u1_struct_0(k5_monoid_0) = k1_numbers
& u1_group_1(k5_monoid_0) = k33_binop_2
& ! [A] :
( m1_subset_1(A,u1_struct_0(k5_monoid_0))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_monoid_0))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> k10_group_1(k5_monoid_0,A,B) = k9_binop_2(C,D) ) ) ) ) ) ) ).
fof(t41_monoid_0,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k5_monoid_0))
<=> m1_subset_1(A,k1_numbers) ) ).
fof(t42_monoid_0,axiom,
k3_binop_1(u1_struct_0(k5_monoid_0),u1_group_1(k5_monoid_0)) = np__0 ).
fof(t43_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& m2_monoid_0(A,k5_monoid_0) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( B = D
& C = E )
=> k10_group_1(A,B,C) = k9_binop_2(D,E) ) ) ) ) ) ) ).
fof(t44_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& m2_monoid_0(A,k5_monoid_0) )
=> k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__0 ) ).
fof(t45_monoid_0,axiom,
$true ).
fof(t46_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& m2_monoid_0(A,k5_monoid_0) )
=> ( A = k6_monoid_0
<=> u1_struct_0(A) = k4_numbers ) ) ).
fof(t47_monoid_0,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k6_monoid_0))
<=> v1_int_1(A) ) ).
fof(d27_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v17_monoid_0(A)
& m5_monoid_0(A,k5_monoid_0,k6_monoid_0) )
=> ( A = k7_monoid_0
<=> u1_struct_0(A) = k5_numbers ) ) ).
fof(d28_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_1(A)
& v1_vectsp_2(A)
& m1_monoid_0(A,k7_monoid_0) )
=> A = k8_monoid_0 ) ).
fof(d29_monoid_0,axiom,
k47_binop_2 = u1_group_1(k7_monoid_0) ).
fof(t48_monoid_0,axiom,
$true ).
fof(t49_monoid_0,axiom,
k7_monoid_0 = g1_group_1(k5_numbers,k47_binop_2) ).
fof(t50_monoid_0,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_monoid_0))
<=> m2_subset_1(A,k1_numbers,k5_numbers) ) ).
fof(t51_monoid_0,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k8_monoid_0))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k8_monoid_0))
=> ( ( A = C
& B = D )
=> k10_group_1(k8_monoid_0,C,D) = k23_binop_2(A,B) ) ) ) ) ) ).
fof(t52_monoid_0,axiom,
k8_monoid_0 = g1_vectsp_1(k5_numbers,k47_binop_2,np__0) ).
fof(t53_monoid_0,axiom,
( k47_binop_2 = k1_realset1(k33_binop_2,k5_numbers)
& k47_binop_2 = k1_realset1(k44_binop_2,k5_numbers) ) ).
fof(t54_monoid_0,axiom,
( r3_binop_1(k5_numbers,np__0,k47_binop_2)
& v9_monoid_0(k47_binop_2,k5_numbers) ) ).
fof(d30_monoid_0,axiom,
k9_monoid_0 = g1_group_1(k1_numbers,k35_binop_2) ).
fof(t55_monoid_0,axiom,
( u1_struct_0(k9_monoid_0) = k1_numbers
& u1_group_1(k9_monoid_0) = k35_binop_2
& ! [A] :
( m1_subset_1(A,u1_struct_0(k9_monoid_0))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k9_monoid_0))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> k10_group_1(k9_monoid_0,A,B) = k11_binop_2(C,D) ) ) ) ) ) ) ).
fof(t56_monoid_0,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k9_monoid_0))
<=> m1_subset_1(A,k1_numbers) ) ).
fof(t57_monoid_0,axiom,
k3_binop_1(u1_struct_0(k9_monoid_0),u1_group_1(k9_monoid_0)) = np__1 ).
fof(t58_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& m2_monoid_0(A,k9_monoid_0) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( B = D
& C = E )
=> k10_group_1(A,B,C) = k11_binop_2(D,E) ) ) ) ) ) ) ).
fof(t59_monoid_0,axiom,
$true ).
fof(t60_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& m2_monoid_0(A,k9_monoid_0) )
=> ( k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__0
| k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__1 ) ) ).
fof(d31_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v17_monoid_0(A)
& m2_monoid_0(A,k9_monoid_0) )
=> ( A = k10_monoid_0
<=> u1_struct_0(A) = k5_numbers ) ) ).
fof(d32_monoid_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_vectsp_1(A)
& v1_vectsp_2(A)
& m1_monoid_0(A,k10_monoid_0) )
=> A = k11_monoid_0 ) ).
fof(d33_monoid_0,axiom,
k48_binop_2 = u1_group_1(k10_monoid_0) ).
fof(t61_monoid_0,axiom,
k10_monoid_0 = g1_group_1(k5_numbers,k48_binop_2) ).
fof(t62_monoid_0,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k10_monoid_0))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k10_monoid_0))
=> ( ( A = C
& B = D )
=> k10_group_1(k10_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) ).
fof(t63_monoid_0,axiom,
k3_binop_1(u1_struct_0(k10_monoid_0),u1_group_1(k10_monoid_0)) = np__1 ).
fof(t64_monoid_0,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k11_monoid_0))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k11_monoid_0))
=> ( ( A = C
& B = D )
=> k10_group_1(k11_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) ).
fof(t65_monoid_0,axiom,
k11_monoid_0 = g1_vectsp_1(k5_numbers,k48_binop_2,np__1) ).
fof(t66_monoid_0,axiom,
k48_binop_2 = k1_realset1(k35_binop_2,k5_numbers) ).
fof(t67_monoid_0,axiom,
( r3_binop_1(k5_numbers,np__1,k48_binop_2)
& v9_monoid_0(k48_binop_2,k5_numbers) ) ).
fof(d34_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v2_monoid_0(B)
& v16_monoid_0(B)
& v17_monoid_0(B)
& l1_group_1(B) )
=> ( B = k12_monoid_0(A)
<=> ( u1_struct_0(B) = k3_finseq_2(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) ) ) ).
fof(d35_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_vectsp_1(B)
& v1_vectsp_2(B)
& m1_monoid_0(B,k12_monoid_0(A)) )
=> B = k13_monoid_0(A) ) ) ).
fof(d36_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k14_monoid_0(A) = u1_group_1(k12_monoid_0(A)) ) ).
fof(t68_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k12_monoid_0(A) = g1_group_1(k3_finseq_2(A),k14_monoid_0(A)) ) ).
fof(t69_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k3_binop_1(u1_struct_0(k12_monoid_0(A)),u1_group_1(k12_monoid_0(A))) = k1_xboole_0 ) ).
fof(t70_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( u1_struct_0(k13_monoid_0(A)) = k3_finseq_2(A)
& u1_group_1(k13_monoid_0(A)) = k14_monoid_0(A)
& u1_vectsp_1(k13_monoid_0(A)) = k1_xboole_0 ) ) ).
fof(t71_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_monoid_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_monoid_0(A)))
=> k4_monoid_0(k13_monoid_0(A),B,C) = k7_finseq_1(B,C) ) ) ) ).
fof(t72_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,k12_monoid_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) ).
fof(t73_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& m2_monoid_0(B,k12_monoid_0(A)) )
=> k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) ).
fof(t74_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,k12_monoid_0(A)) )
=> ( m1_subset_1(k1_xboole_0,u1_struct_0(B))
=> ( v2_group_1(B)
& k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) ) ) ).
fof(t75_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r1_tarski(A,B)
=> m2_monoid_0(k12_monoid_0(A),k12_monoid_0(B)) ) ) ) ).
fof(t76_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_setwiseo(k14_monoid_0(A),k3_finseq_2(A))
& k3_binop_1(k3_finseq_2(A),k14_monoid_0(A)) = k1_xboole_0
& v2_binop_1(k14_monoid_0(A),k3_finseq_2(A)) ) ) ).
fof(d37_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v1_monoid_0(B)
& l1_group_1(B) )
=> ( B = k15_monoid_0(A)
<=> ( u1_struct_0(B) = k4_partfun1(A,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) ) ) ).
fof(d38_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v1_vectsp_1(B)
& v1_vectsp_2(B)
& m1_monoid_0(B,k15_monoid_0(A)) )
=> B = k16_monoid_0(A) ) ).
fof(d39_monoid_0,axiom,
! [A] : k17_monoid_0(A) = u1_group_1(k15_monoid_0(A)) ).
fof(t77_monoid_0,axiom,
! [A,B] :
( m1_subset_1(A,u1_struct_0(k15_monoid_0(B)))
<=> ( v1_funct_1(A)
& m2_relset_1(A,B,B) ) ) ).
fof(t78_monoid_0,axiom,
! [A] : k3_binop_1(u1_struct_0(k15_monoid_0(A)),u1_group_1(k15_monoid_0(A))) = k6_partfun1(A) ).
fof(t79_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,k15_monoid_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) ).
fof(t80_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& m2_monoid_0(B,k15_monoid_0(A)) )
=> ( m1_subset_1(k6_partfun1(A),u1_struct_0(B))
=> ( v2_group_1(B)
& k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k6_partfun1(A) ) ) ) ).
fof(t81_monoid_0,axiom,
! [A,B] :
( r1_tarski(A,B)
=> m2_monoid_0(k15_monoid_0(A),k15_monoid_0(B)) ) ).
fof(d40_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& m2_monoid_0(B,k15_monoid_0(A)) )
=> ( B = k18_monoid_0(A)
<=> u1_struct_0(B) = k1_funct_2(A,A) ) ) ).
fof(d41_monoid_0,axiom,
! [A,B] :
( ( v1_vectsp_1(B)
& v1_vectsp_2(B)
& m1_monoid_0(B,k18_monoid_0(A)) )
=> B = k19_monoid_0(A) ) ).
fof(t82_monoid_0,axiom,
! [A,B] :
( m1_subset_1(A,u1_struct_0(k18_monoid_0(B)))
<=> ( v1_funct_1(A)
& v1_funct_2(A,B,B)
& m2_relset_1(A,B,B) ) ) ).
fof(t83_monoid_0,axiom,
! [A] : u1_group_1(k18_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A)) ).
fof(t84_monoid_0,axiom,
! [A] : k3_binop_1(u1_struct_0(k18_monoid_0(A)),u1_group_1(k18_monoid_0(A))) = k6_partfun1(A) ).
fof(t85_monoid_0,axiom,
! [A] :
( u1_struct_0(k19_monoid_0(A)) = k1_funct_2(A,A)
& u1_group_1(k19_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A))
& u1_vectsp_1(k19_monoid_0(A)) = k6_partfun1(A) ) ).
fof(d42_monoid_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v13_monoid_0(B)
& m5_monoid_0(B,k15_monoid_0(A),k18_monoid_0(A)) )
=> ( B = k20_monoid_0(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k18_monoid_0(A)))
=> ( r2_hidden(C,u1_struct_0(B))
<=> ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) ) ) ) ) ) ).
fof(t86_monoid_0,axiom,
! [A,B] :
( m1_subset_1(A,u1_struct_0(k20_monoid_0(B)))
<=> ( v1_funct_1(A)
& v1_funct_2(A,B,B)
& v3_funct_2(A,B,B)
& m2_relset_1(A,B,B) ) ) ).
fof(t87_monoid_0,axiom,
! [A] :
( k3_binop_1(u1_struct_0(k20_monoid_0(A)),u1_group_1(k20_monoid_0(A))) = k6_partfun1(A)
& k2_group_1(k20_monoid_0(A)) = k6_partfun1(A) ) ).
fof(t88_monoid_0,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k20_monoid_0(A)))
=> k3_group_1(k20_monoid_0(A),B) = k2_funct_1(B) ) ).
fof(t89_monoid_0,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_fraenkel(u1_struct_0(A))
=> v1_monoid_0(A) ) ) ).
fof(s1_monoid_0,axiom,
( ! [A] :
( m1_struct_0(A,f1_s1_monoid_0,f2_s1_monoid_0)
=> ! [B] :
( m1_struct_0(B,f1_s1_monoid_0,f2_s1_monoid_0)
=> r2_hidden(k1_group_1(f1_s1_monoid_0,A,B),f2_s1_monoid_0) ) )
=> ? [A] :
( ~ v3_struct_0(A)
& v1_group_1(A)
& m2_monoid_0(A,f1_s1_monoid_0)
& u1_struct_0(A) = f2_s1_monoid_0 ) ) ).
fof(s2_monoid_0,axiom,
( ( ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
=> ( ( p1_s2_monoid_0(A)
& p1_s2_monoid_0(B) )
=> p1_s2_monoid_0(k1_group_1(f1_s2_monoid_0,A,B)) ) ) )
& ? [A] :
( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
& p1_s2_monoid_0(A) ) )
=> ? [A] :
( ~ v3_struct_0(A)
& v1_group_1(A)
& m2_monoid_0(A,f1_s2_monoid_0)
& ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
=> ( r2_hidden(B,u1_struct_0(A))
<=> p1_s2_monoid_0(B) ) ) ) ) ).
fof(s3_monoid_0,axiom,
( ( ! [A] :
( m1_struct_0(A,f1_s3_monoid_0,f2_s3_monoid_0)
=> ! [B] :
( m1_struct_0(B,f1_s3_monoid_0,f2_s3_monoid_0)
=> r2_hidden(k1_group_1(f1_s3_monoid_0,A,B),f2_s3_monoid_0) ) )
& r2_hidden(u1_vectsp_1(f1_s3_monoid_0),f2_s3_monoid_0) )
=> ? [A] :
( ~ v3_struct_0(A)
& v1_vectsp_1(A)
& m3_monoid_0(A,f1_s3_monoid_0)
& u1_struct_0(A) = f2_s3_monoid_0 ) ) ).
fof(s4_monoid_0,axiom,
( ( ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s4_monoid_0))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
=> ( ( p1_s4_monoid_0(A)
& p1_s4_monoid_0(B) )
=> p1_s4_monoid_0(k1_group_1(f1_s4_monoid_0,A,B)) ) ) )
& p1_s4_monoid_0(u1_vectsp_1(f1_s4_monoid_0)) )
=> ? [A] :
( ~ v3_struct_0(A)
& v1_vectsp_1(A)
& m3_monoid_0(A,f1_s4_monoid_0)
& ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
=> ( r2_hidden(B,u1_struct_0(A))
<=> p1_s4_monoid_0(B) ) ) ) ) ).
fof(dt_m1_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( m1_monoid_0(B,A)
=> l1_vectsp_1(B) ) ) ).
fof(existence_m1_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] : m1_monoid_0(B,A) ) ).
fof(dt_m2_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( m2_monoid_0(B,A)
=> l1_group_1(B) ) ) ).
fof(existence_m2_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] : m2_monoid_0(B,A) ) ).
fof(dt_m3_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ! [B] :
( m3_monoid_0(B,A)
=> l1_vectsp_1(B) ) ) ).
fof(existence_m3_monoid_0,axiom,
! [A] :
( l1_group_1(A)
=> ? [B] : m3_monoid_0(B,A) ) ).
fof(dt_m4_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m1_monoid_0(B,A) )
=> ! [C] :
( m4_monoid_0(C,A,B)
=> m2_monoid_0(C,A) ) ) ).
fof(existence_m4_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m1_monoid_0(B,A) )
=> ? [C] : m4_monoid_0(C,A,B) ) ).
fof(redefinition_m4_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m1_monoid_0(B,A) )
=> ! [C] :
( m4_monoid_0(C,A,B)
<=> m2_monoid_0(C,B) ) ) ).
fof(dt_m5_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m2_monoid_0(B,A) )
=> ! [C] :
( m5_monoid_0(C,A,B)
=> m2_monoid_0(C,A) ) ) ).
fof(existence_m5_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m2_monoid_0(B,A) )
=> ? [C] : m5_monoid_0(C,A,B) ) ).
fof(redefinition_m5_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m2_monoid_0(B,A) )
=> ! [C] :
( m5_monoid_0(C,A,B)
<=> m2_monoid_0(C,B) ) ) ).
fof(dt_m6_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ! [C] :
( m6_monoid_0(C,A,B)
=> m2_monoid_0(C,A) ) ) ).
fof(existence_m6_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ? [C] : m6_monoid_0(C,A,B) ) ).
fof(redefinition_m6_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ! [C] :
( m6_monoid_0(C,A,B)
<=> m2_monoid_0(C,B) ) ) ).
fof(dt_m7_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ! [C] :
( m7_monoid_0(C,A,B)
=> m3_monoid_0(C,A) ) ) ).
fof(existence_m7_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ? [C] : m7_monoid_0(C,A,B) ) ).
fof(redefinition_m7_monoid_0,axiom,
! [A,B] :
( ( l1_group_1(A)
& m3_monoid_0(B,A) )
=> ! [C] :
( m7_monoid_0(C,A,B)
<=> m3_monoid_0(C,B) ) ) ).
fof(dt_k1_monoid_0,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,A)
& m1_finseq_1(C,A) )
=> m2_finseq_2(k1_monoid_0(A,B,C),A,k3_finseq_2(A)) ) ).
fof(redefinition_k1_monoid_0,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,A)
& m1_finseq_1(C,A) )
=> k1_monoid_0(A,B,C) = k7_finseq_1(B,C) ) ).
fof(dt_k2_monoid_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v1_funct_1(k2_monoid_0(A,B,C))
& v1_funct_2(k2_monoid_0(A,B,C),A,A)
& m2_relset_1(k2_monoid_0(A,B,C),A,A) ) ) ).
fof(redefinition_k2_monoid_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> k2_monoid_0(A,B,C) = k5_relat_1(B,C) ) ).
fof(dt_k3_monoid_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& v3_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v1_funct_1(k3_monoid_0(A,B,C))
& v1_funct_2(k3_monoid_0(A,B,C),A,A)
& v3_funct_2(k3_monoid_0(A,B,C),A,A)
& m2_relset_1(k3_monoid_0(A,B,C),A,A) ) ) ).
fof(redefinition_k3_monoid_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& v3_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> k3_monoid_0(A,B,C) = k5_relat_1(B,C) ) ).
fof(dt_k4_monoid_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_monoid_0(A,B,C),u1_struct_0(A)) ) ).
fof(redefinition_k4_monoid_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k4_monoid_0(A,B,C) = k1_group_1(A,B,C) ) ).
fof(dt_k5_monoid_0,axiom,
( ~ v3_struct_0(k5_monoid_0)
& v1_group_1(k5_monoid_0)
& v2_group_1(k5_monoid_0)
& v4_group_1(k5_monoid_0)
& v7_group_1(k5_monoid_0)
& v13_monoid_0(k5_monoid_0)
& v16_monoid_0(k5_monoid_0)
& l1_group_1(k5_monoid_0) ) ).
fof(dt_k6_monoid_0,axiom,
( ~ v3_struct_0(k6_monoid_0)
& v1_group_1(k6_monoid_0)
& v2_group_1(k6_monoid_0)
& v13_monoid_0(k6_monoid_0)
& m2_monoid_0(k6_monoid_0,k5_monoid_0) ) ).
fof(redefinition_k6_monoid_0,axiom,
k6_monoid_0 = k3_gr_cy_1 ).
fof(dt_k7_monoid_0,axiom,
( ~ v3_struct_0(k7_monoid_0)
& v1_group_1(k7_monoid_0)
& v2_group_1(k7_monoid_0)
& v17_monoid_0(k7_monoid_0)
& m5_monoid_0(k7_monoid_0,k5_monoid_0,k6_monoid_0) ) ).
fof(dt_k8_monoid_0,axiom,
( ~ v3_struct_0(k8_monoid_0)
& v1_vectsp_1(k8_monoid_0)
& v1_vectsp_2(k8_monoid_0)
& m1_monoid_0(k8_monoid_0,k7_monoid_0) ) ).
fof(dt_k9_monoid_0,axiom,
( ~ v3_struct_0(k9_monoid_0)
& v1_group_1(k9_monoid_0)
& v2_group_1(k9_monoid_0)
& v4_group_1(k9_monoid_0)
& v7_group_1(k9_monoid_0)
& l1_group_1(k9_monoid_0) ) ).
fof(dt_k10_monoid_0,axiom,
( ~ v3_struct_0(k10_monoid_0)
& v1_group_1(k10_monoid_0)
& v2_group_1(k10_monoid_0)
& v17_monoid_0(k10_monoid_0)
& m2_monoid_0(k10_monoid_0,k9_monoid_0) ) ).
fof(dt_k11_monoid_0,axiom,
( ~ v3_struct_0(k11_monoid_0)
& v1_vectsp_1(k11_monoid_0)
& v1_vectsp_2(k11_monoid_0)
& m1_monoid_0(k11_monoid_0,k10_monoid_0) ) ).
fof(dt_k12_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k12_monoid_0(A))
& v1_group_1(k12_monoid_0(A))
& v2_group_1(k12_monoid_0(A))
& v4_group_1(k12_monoid_0(A))
& v2_monoid_0(k12_monoid_0(A))
& v16_monoid_0(k12_monoid_0(A))
& v17_monoid_0(k12_monoid_0(A))
& l1_group_1(k12_monoid_0(A)) ) ) ).
fof(dt_k13_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k13_monoid_0(A))
& v1_vectsp_1(k13_monoid_0(A))
& v1_vectsp_2(k13_monoid_0(A))
& m1_monoid_0(k13_monoid_0(A),k12_monoid_0(A)) ) ) ).
fof(dt_k14_monoid_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k14_monoid_0(A))
& v1_funct_2(k14_monoid_0(A),k2_zfmisc_1(k3_finseq_2(A),k3_finseq_2(A)),k3_finseq_2(A))
& m2_relset_1(k14_monoid_0(A),k2_zfmisc_1(k3_finseq_2(A),k3_finseq_2(A)),k3_finseq_2(A)) ) ) ).
fof(dt_k15_monoid_0,axiom,
! [A] :
( ~ v3_struct_0(k15_monoid_0(A))
& v1_group_1(k15_monoid_0(A))
& v2_group_1(k15_monoid_0(A))
& v4_group_1(k15_monoid_0(A))
& v1_monoid_0(k15_monoid_0(A))
& l1_group_1(k15_monoid_0(A)) ) ).
fof(dt_k16_monoid_0,axiom,
! [A] :
( ~ v3_struct_0(k16_monoid_0(A))
& v1_vectsp_1(k16_monoid_0(A))
& v1_vectsp_2(k16_monoid_0(A))
& m1_monoid_0(k16_monoid_0(A),k15_monoid_0(A)) ) ).
fof(dt_k17_monoid_0,axiom,
! [A] :
( v1_funct_1(k17_monoid_0(A))
& v1_funct_2(k17_monoid_0(A),k2_zfmisc_1(k4_partfun1(A,A),k4_partfun1(A,A)),k4_partfun1(A,A))
& m2_relset_1(k17_monoid_0(A),k2_zfmisc_1(k4_partfun1(A,A),k4_partfun1(A,A)),k4_partfun1(A,A)) ) ).
fof(dt_k18_monoid_0,axiom,
! [A] :
( ~ v3_struct_0(k18_monoid_0(A))
& v1_group_1(k18_monoid_0(A))
& v2_group_1(k18_monoid_0(A))
& m2_monoid_0(k18_monoid_0(A),k15_monoid_0(A)) ) ).
fof(dt_k19_monoid_0,axiom,
! [A] :
( v1_vectsp_1(k19_monoid_0(A))
& v1_vectsp_2(k19_monoid_0(A))
& m1_monoid_0(k19_monoid_0(A),k18_monoid_0(A)) ) ).
fof(dt_k20_monoid_0,axiom,
! [A] :
( ~ v3_struct_0(k20_monoid_0(A))
& v1_group_1(k20_monoid_0(A))
& v2_group_1(k20_monoid_0(A))
& v13_monoid_0(k20_monoid_0(A))
& m5_monoid_0(k20_monoid_0(A),k15_monoid_0(A),k18_monoid_0(A)) ) ).
%------------------------------------------------------------------------------