TPTP Problem File: GRP651+1.p
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%------------------------------------------------------------------------------
% File : GRP651+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Group Theory
% Problem : On the Lattice of Subgroups of a Group T34
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Gan96] Ganczarski (1996), On the Lattice of Subgroups of a Gr
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t34_latsubgr [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 102 ( 25 unt; 0 def)
% Number of atoms : 475 ( 42 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 438 ( 65 ~; 2 |; 233 &)
% ( 15 <=>; 123 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 37 ( 35 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 1 con; 0-4 aty)
% Number of variables : 220 ( 200 !; 20 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t34_latsubgr,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_funct_1(C)
=> v2_funct_1(k3_latsubgr(A,B,C)) ) ) ) ) ).
fof(abstractness_v1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> ( v1_group_1(A)
=> A = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ).
fof(abstractness_v3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( v3_lattices(A)
=> A = g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_funct_2,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_partfun1(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,A,B) ) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(cc5_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& ~ v1_xboole_0(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( r1_tarski(A,B)
& r1_tarski(B,A) ) ) ).
fof(d12_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( C = k9_relat_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( r2_hidden(E,k1_relat_1(A))
& r2_hidden(E,B)
& D = k1_funct_1(A,E) ) ) ) ) ).
fof(d1_funct_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> ( v1_funct_2(C,A,B)
<=> A = k4_relset_1(A,B,C) ) )
& ( B = k1_xboole_0
=> ( A = k1_xboole_0
| ( v1_funct_2(C,A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ).
fof(d1_group_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( B = k1_group_3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_group_1(C)
& m1_group_2(C,A) ) ) ) ) ).
fof(d1_rlvect_1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( r1_rlvect_1(A,B)
<=> r2_hidden(B,u1_struct_0(A)) ) ) ).
fof(d3_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_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)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B)))
& m2_relset_1(D,u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B))) )
=> ( D = k3_latsubgr(A,B,C)
<=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ( F = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
=> k1_funct_1(D,E) = k5_group_4(B,F) ) ) ) ) ) ) ) ) ).
fof(d5_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& l1_group_1(B) )
=> ( m1_group_2(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) ) ) ).
fof(d7_group_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_group_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_group_6(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_group_1(A,D,E)) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(d8_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A))
& k1_funct_1(A,B) = k1_funct_1(A,C) )
=> B = C ) ) ) ).
fof(dt_g1_group_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_group_1(g1_group_1(A,B))
& l1_group_1(g1_group_1(A,B)) ) ) ).
fof(dt_g3_lattices,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v3_lattices(g3_lattices(A,B,C))
& l3_lattices(g3_lattices(A,B,C)) ) ) ).
fof(dt_k11_group_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k11_group_4(A))
& v3_lattices(k11_group_4(A))
& v10_lattices(k11_group_4(A))
& l3_lattices(k11_group_4(A)) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_group_1,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(k1_group_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k1_group_3,axiom,
$true ).
fof(dt_k1_realset1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(dt_k2_group_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_group_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_group_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k3_latsubgr,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k3_latsubgr(A,B,C))
& v1_funct_2(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B)))
& m2_relset_1(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B))) ) ) ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_k5_group_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_group_1(k5_group_4(A,B))
& m1_group_2(k5_group_4(A,B),A) ) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k8_funct_2(A,B,C,D),B) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_l1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( l1_lattices(A)
& l2_lattices(A) ) ) ).
fof(dt_m1_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ( ~ v3_struct_0(B)
& v3_group_1(B)
& l1_group_1(B) ) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_u1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> ( v1_funct_1(u1_group_1(A))
& v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_u1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> ( v1_funct_1(u1_lattices(A))
& v1_funct_2(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> ( v1_funct_1(u2_lattices(A))
& v1_funct_2(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(existence_l1_group_1,axiom,
? [A] : l1_group_1(A) ).
fof(existence_l1_lattices,axiom,
? [A] : l1_lattices(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l2_lattices,axiom,
? [A] : l2_lattices(A) ).
fof(existence_l3_lattices,axiom,
? [A] : l3_lattices(A) ).
fof(existence_m1_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ? [B] : m1_group_2(B,A) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(fc1_group_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ~ v1_xboole_0(k1_group_3(A)) ) ).
fof(fc1_group_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k11_group_4(A))
& v3_lattices(k11_group_4(A))
& v4_lattices(k11_group_4(A))
& v5_lattices(k11_group_4(A))
& v6_lattices(k11_group_4(A))
& v7_lattices(k11_group_4(A))
& v8_lattices(k11_group_4(A))
& v9_lattices(k11_group_4(A))
& v10_lattices(k11_group_4(A))
& v13_lattices(k11_group_4(A))
& v14_lattices(k11_group_4(A))
& v15_lattices(k11_group_4(A)) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(free_g1_group_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g1_group_1(A,B) = g1_group_1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(free_g3_lattices,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D,E,F] :
( g3_lattices(A,B,C) = g3_lattices(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B) ) ).
fof(rc1_partfun1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_xboole_0(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(redefinition_k2_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k4_relset_1(A,B,C) = k1_relat_1(C) ) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(redefinition_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> ( r1_group_2(A,B,C)
<=> B = C ) ) ).
fof(reflexivity_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> r1_group_2(A,B,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(symmetry_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> ( r1_group_2(A,B,C)
=> r1_group_2(A,C,B) ) ) ).
fof(t116_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
~ ( r2_hidden(E,k2_funct_2(A,B,D,C))
& ! [F] :
( m1_subset_1(F,A)
=> ~ ( r2_hidden(F,C)
& E = k1_funct_1(D,F) ) ) ) ) ).
fof(t157_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B))
& r1_tarski(A,k1_relat_1(C))
& v2_funct_1(C) )
=> r1_tarski(A,B) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( B = u1_struct_0(C)
=> r1_group_2(A,k5_group_4(A,B),C) ) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t41_group_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(D,A,B)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,k3_group_1(A,C)) = k3_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)) ) ) ) ) ).
fof(t43_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( B != k1_xboole_0
=> ! [E] :
( ? [F] :
( r2_hidden(F,A)
& r2_hidden(F,C)
& E = k1_funct_1(D,F) )
=> r2_hidden(E,k9_relat_1(D,C)) ) ) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t55_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> r1_rlvect_1(B,k2_group_1(A)) ) ) ).
fof(t59_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_group_2(D,A)
=> ( ( r1_rlvect_1(D,B)
& r1_rlvect_1(D,C) )
=> r1_rlvect_1(D,k1_group_1(A,B,C)) ) ) ) ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t60_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_group_2(C,A)
=> ( r1_rlvect_1(C,B)
=> r1_rlvect_1(C,k3_group_1(A,B)) ) ) ) ) ).
fof(t61_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k1_group_1(A,C,D),B) ) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k3_group_1(A,C),B) ) )
& ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> u1_struct_0(C) != B ) ) ) ) ).
fof(t68_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(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(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(t92_group_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> u1_struct_0(k11_group_4(A)) = k1_group_3(A) ) ).
%------------------------------------------------------------------------------