SET007 Axioms: SET007+343.ax
%------------------------------------------------------------------------------
% File : SET007+343 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Opposite Rings, Modules and Their Morphisms
% 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 : mod_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 110 ( 4 unt; 0 def)
% Number of atoms : 1735 ( 106 equ)
% Maximal formula atoms : 47 ( 15 avg)
% Number of connectives : 1864 ( 239 ~; 0 |;1252 &)
% ( 46 <=>; 327 =>; 0 <=; 0 <~>)
% Maximal formula depth : 43 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 59 ( 57 usr; 1 prp; 0-6 aty)
% Number of functors : 37 ( 37 usr; 0 con; 1-6 aty)
% Number of variables : 351 ( 348 !; 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_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc2_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v2_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc3_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc4_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v2_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v4_vectsp_1(k2_mod_4(A))
& v5_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc5_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v2_group_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v4_vectsp_1(k2_mod_4(A))
& v5_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc6_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v2_group_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v4_vectsp_1(k2_mod_4(A))
& v5_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& v9_vectsp_1(k2_mod_4(A))
& ~ v10_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc7_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v2_group_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v4_vectsp_1(k2_mod_4(A))
& v5_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& v9_vectsp_1(k2_mod_4(A))
& ~ v10_vectsp_1(k2_mod_4(A)) ) ) ).
fof(fc8_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ( ~ v3_struct_0(k3_mod_4(A,B))
& v3_vectsp_2(k3_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(fc9_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(k5_mod_4(A,B))
& v11_vectsp_1(k5_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(fc10_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ( ~ v3_struct_0(k3_mod_4(A,B))
& v3_rlvect_1(k3_mod_4(A,B))
& v4_rlvect_1(k3_mod_4(A,B))
& v5_rlvect_1(k3_mod_4(A,B))
& v6_rlvect_1(k3_mod_4(A,B))
& v3_vectsp_2(k3_mod_4(A,B),k2_mod_4(A))
& v5_vectsp_2(k3_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(fc11_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(k5_mod_4(A,B))
& v3_rlvect_1(k5_mod_4(A,B))
& v4_rlvect_1(k5_mod_4(A,B))
& v5_rlvect_1(k5_mod_4(A,B))
& v6_rlvect_1(k5_mod_4(A,B))
& v11_vectsp_1(k5_mod_4(A,B),k2_mod_4(A))
& v12_vectsp_1(k5_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(rc1_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ? [B] :
( m1_mod_4(B,A,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_mod_4(B,A,A) ) ) ).
fof(t1_mod_4,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(B,C),A)
& m2_relset_1(D,k2_zfmisc_1(B,C),A) )
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,C)
=> k2_binop_1(B,C,A,D,E,F) = k2_binop_1(C,B,A,k1_mod_4(B,C,A,D),F,E) ) ) ) ) ) ) ).
fof(d1_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> k2_mod_4(A) = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),k1_mod_4(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A)),u1_vectsp_1(A),u2_struct_0(A)) ) ).
fof(t2_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( g1_rlvect_1(u1_struct_0(k2_mod_4(A)),u1_rlvect_1(k2_mod_4(A)),u2_struct_0(k2_mod_4(A))) = g1_rlvect_1(u1_struct_0(A),u1_rlvect_1(A),u2_struct_0(A))
& ( ( v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A) )
=> k7_vectsp_1(k2_mod_4(A)) = k7_vectsp_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(k2_mod_4(A)))
<=> m1_subset_1(B,u1_struct_0(A)) ) ) ) ).
fof(t3_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> k2_mod_4(k2_mod_4(A)) = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ).
fof(t4_mod_4,axiom,
( ! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> k2_group_1(A) = k2_group_1(k2_mod_4(A)) )
& ! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( k1_rlvect_1(A) = k1_rlvect_1(k2_mod_4(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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k2_mod_4(A)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k2_mod_4(A)))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k2_mod_4(A)))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k2_mod_4(A)))
=> ( ( B = F
& C = G
& D = H
& E = I )
=> ( k2_rlvect_1(A,B,C) = k2_rlvect_1(k2_mod_4(A),F,G)
& k1_group_1(A,B,C) = k1_group_1(k2_mod_4(A),G,F)
& k5_rlvect_1(A,B) = k5_rlvect_1(k2_mod_4(A),F)
& k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(k2_mod_4(A),k2_rlvect_1(k2_mod_4(A),F,G),H)
& k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(k2_mod_4(A),F,k2_rlvect_1(k2_mod_4(A),G,H))
& k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(k2_mod_4(A),H,k1_group_1(k2_mod_4(A),G,F))
& k1_group_1(A,B,k1_group_1(A,C,D)) = k1_group_1(k2_mod_4(A),k1_group_1(k2_mod_4(A),H,G),F)
& k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k1_group_1(k2_mod_4(A),k2_rlvect_1(k2_mod_4(A),G,H),F)
& k1_group_1(A,k2_rlvect_1(A,C,D),B) = k1_group_1(k2_mod_4(A),F,k2_rlvect_1(k2_mod_4(A),G,H))
& k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,D,E)) = k2_rlvect_1(k2_mod_4(A),k1_group_1(k2_mod_4(A),G,F),k1_group_1(k2_mod_4(A),I,H)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t5_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& l3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(t6_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& l3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(t7_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v6_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& v9_vectsp_1(k2_mod_4(A))
& ~ v10_vectsp_1(k2_mod_4(A))
& l3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(t8_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k2_mod_4(A))
& v3_rlvect_1(k2_mod_4(A))
& v4_rlvect_1(k2_mod_4(A))
& v5_rlvect_1(k2_mod_4(A))
& v6_rlvect_1(k2_mod_4(A))
& v4_group_1(k2_mod_4(A))
& v7_group_1(k2_mod_4(A))
& v3_vectsp_1(k2_mod_4(A))
& v7_vectsp_1(k2_mod_4(A))
& v8_vectsp_1(k2_mod_4(A))
& v9_vectsp_1(k2_mod_4(A))
& ~ v10_vectsp_1(k2_mod_4(A))
& l3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(d2_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,k2_mod_4(A))
& l1_vectsp_2(C,k2_mod_4(A)) )
=> ( C = k3_mod_4(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(k2_mod_4(A))),u1_struct_0(B))
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(k2_mod_4(A))),u1_struct_0(B)) )
=> ( D = k1_mod_4(u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),u2_vectsp_1(A,B))
=> C = g1_vectsp_2(k2_mod_4(A),u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),D) ) ) ) ) ) ) ).
fof(t9_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ( g1_rlvect_1(u1_struct_0(k3_mod_4(A,B)),u1_rlvect_1(k3_mod_4(A,B)),u2_struct_0(k3_mod_4(A,B))) = g1_rlvect_1(u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B))
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
<=> m1_subset_1(C,u1_struct_0(k3_mod_4(A,B))) ) ) ) ) ).
fof(d3_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B)) )
=> k4_mod_4(A,B,C) = k1_mod_4(u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),C) ) ) ) ).
fof(t10_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> u1_vectsp_2(k2_mod_4(A),k3_mod_4(A,B)) = k4_mod_4(A,B,u2_vectsp_1(A,B)) ) ) ).
fof(d4_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v11_vectsp_1(C,k2_mod_4(A))
& l4_vectsp_1(C,k2_mod_4(A)) )
=> ( C = k5_mod_4(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(k2_mod_4(A)),u1_struct_0(B)),u1_struct_0(B))
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(k2_mod_4(A)),u1_struct_0(B)),u1_struct_0(B)) )
=> ( D = k1_mod_4(u1_struct_0(B),u1_struct_0(A),u1_struct_0(B),u1_vectsp_2(A,B))
=> C = g4_vectsp_1(k2_mod_4(A),u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),D) ) ) ) ) ) ) ).
fof(t11_mod_4,axiom,
$true ).
fof(t12_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ( g1_rlvect_1(u1_struct_0(k5_mod_4(A,B)),u1_rlvect_1(k5_mod_4(A,B)),u2_struct_0(k5_mod_4(A,B))) = g1_rlvect_1(u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B))
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
<=> m1_subset_1(C,u1_struct_0(k5_mod_4(A,B))) ) ) ) ) ).
fof(d5_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B)) )
=> k6_mod_4(A,B,C) = k1_mod_4(u1_struct_0(B),u1_struct_0(A),u1_struct_0(B),C) ) ) ) ).
fof(t13_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> u2_vectsp_1(k2_mod_4(A),k5_mod_4(A,B)) = k6_mod_4(A,B,u1_vectsp_2(A,B)) ) ) ).
fof(t14_mod_4,axiom,
$true ).
fof(t15_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B)) )
=> k6_mod_4(k2_mod_4(A),k3_mod_4(A,B),k4_mod_4(A,B,C)) = C ) ) ) ).
fof(t16_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_mod_4(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k3_mod_4(A,B)))
=> ( ( D = E
& F = G )
=> k2_binop_1(u1_struct_0(k3_mod_4(A,B)),u1_struct_0(k2_mod_4(A)),u1_struct_0(k3_mod_4(A,B)),k4_mod_4(A,B,C),G,E) = k2_binop_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),C,D,F) ) ) ) ) ) ) ) ) ).
fof(t17_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& l1_vectsp_2(D,B) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> ( ( B = k2_mod_4(A)
& D = k3_mod_4(A,C)
& E = F
& G = H )
=> k6_vectsp_2(B,D,F,H) = k6_vectsp_1(A,C,E,G) ) ) ) ) ) ) ) ) ) ).
fof(t18_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& l1_vectsp_2(D,B) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(D))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> ( ( B = k2_mod_4(A)
& D = k3_mod_4(A,C)
& E = G
& F = H )
=> k2_rlvect_1(D,G,H) = k2_rlvect_1(C,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t19_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B)) )
=> k4_mod_4(k2_mod_4(A),k5_mod_4(A,B),k6_mod_4(A,B,C)) = C ) ) ) ).
fof(t20_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_mod_4(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k5_mod_4(A,B)))
=> ( ( D = E
& F = G )
=> k2_binop_1(u1_struct_0(k2_mod_4(A)),u1_struct_0(k5_mod_4(A,B)),u1_struct_0(k5_mod_4(A,B)),k6_mod_4(A,B,C),E,G) = k2_binop_1(u1_struct_0(B),u1_struct_0(A),u1_struct_0(B),C,F,D) ) ) ) ) ) ) ) ) ).
fof(t21_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& l1_vectsp_2(D,B) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> ( ( A = k2_mod_4(B)
& C = k5_mod_4(B,D)
& E = F
& G = H )
=> k6_vectsp_2(B,D,F,H) = k6_vectsp_1(A,C,E,G) ) ) ) ) ) ) ) ) ) ).
fof(t22_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& l1_vectsp_2(D,B) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(D))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> ( ( A = k2_mod_4(B)
& C = k5_mod_4(B,D)
& E = G
& F = H )
=> k2_rlvect_1(D,G,H) = k2_rlvect_1(C,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t23_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> k5_mod_4(k2_mod_4(A),k3_mod_4(A,B)) = g4_vectsp_1(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u2_vectsp_1(A,B)) ) ) ).
fof(t24_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> k3_mod_4(k2_mod_4(A),k5_mod_4(A,B)) = g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B)) ) ) ).
fof(t25_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ( ~ v3_struct_0(k3_mod_4(A,B))
& v3_rlvect_1(k3_mod_4(A,B))
& v4_rlvect_1(k3_mod_4(A,B))
& v5_rlvect_1(k3_mod_4(A,B))
& v6_rlvect_1(k3_mod_4(A,B))
& v3_vectsp_2(k3_mod_4(A,B),k2_mod_4(A))
& v5_vectsp_2(k3_mod_4(A,B),k2_mod_4(A))
& l1_vectsp_2(k3_mod_4(A,B),k2_mod_4(A)) ) ) ) ).
fof(t26_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(k5_mod_4(A,B))
& v3_rlvect_1(k5_mod_4(A,B))
& v4_rlvect_1(k5_mod_4(A,B))
& v5_rlvect_1(k5_mod_4(A,B))
& v6_rlvect_1(k5_mod_4(A,B))
& v11_vectsp_1(k5_mod_4(A,B),k2_mod_4(A))
& v12_vectsp_1(k5_mod_4(A,B),k2_mod_4(A))
& l4_vectsp_1(k5_mod_4(A,B),k2_mod_4(A)) ) ) ) ).
fof(d6_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_mod_4(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,k2_rlvect_1(A,D,E)) = k2_rlvect_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)) ) )
& ! [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,E),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,k2_group_1(A)) = k2_group_1(B) ) ) ) ) ) ).
fof(d7_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_mod_4(C,A,B)
<=> ( v1_ringcat1(C,A,B)
& v2_funct_1(C) ) ) ) ) ) ).
fof(d8_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_mod_4(C,A,B)
<=> ( v1_mod_4(C,A,B)
& v2_funct_1(C) ) ) ) ) ) ).
fof(d9_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v4_mod_4(C,A,B)
<=> ( v1_ringcat1(C,A,B)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d10_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_mod_4(C,A,B)
<=> ( v1_mod_4(C,A,B)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d11_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v6_mod_4(C,A,B)
<=> ( v2_mod_4(C,A,B)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d12_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v7_mod_4(C,A,B)
<=> ( v3_mod_4(C,A,B)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d13_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v8_mod_4(B,A)
<=> v1_ringcat1(B,A,A) ) ) ) ).
fof(d14_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v9_mod_4(B,A)
<=> v1_mod_4(B,A,A) ) ) ) ).
fof(d15_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v10_mod_4(B,A)
<=> v6_mod_4(B,A,A) ) ) ) ).
fof(d16_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v11_mod_4(B,A)
<=> v7_mod_4(B,A,A) ) ) ) ).
fof(t27_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v10_mod_4(B,A)
<=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k1_group_1(A,C,D)) = k1_group_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) )
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k2_group_1(A)) = k2_group_1(A)
& v2_funct_1(B)
& k2_relat_1(B) = u1_struct_0(A) ) ) ) ) ).
fof(t28_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v11_mod_4(B,A)
<=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k1_group_1(A,C,D)) = k1_group_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C)) ) )
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k2_group_1(A)) = k2_group_1(A)
& v2_funct_1(B)
& k2_relat_1(B) = u1_struct_0(A) ) ) ) ) ).
fof(t29_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> v10_mod_4(k7_grcat_1(A),A) ) ).
fof(t30_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( v1_ringcat1(C,A,B)
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_rlvect_1(A)) = k1_rlvect_1(B)
& k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k5_rlvect_1(A,D)) = k5_rlvect_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,k6_rlvect_1(A,D,E)) = k6_rlvect_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(t31_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( v1_mod_4(C,A,B)
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_rlvect_1(A)) = k1_rlvect_1(B)
& k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k5_rlvect_1(A,D)) = k5_rlvect_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,k6_rlvect_1(A,D,E)) = k6_rlvect_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(t32_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( v11_mod_4(k7_grcat_1(A),A)
<=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) ) ) ) ).
fof(t33_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( v11_mod_4(k7_grcat_1(A),A)
<=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) ) ) ) ).
fof(d17_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> k7_mod_4(A,B,C) = C ) ) ) ).
fof(t34_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> k7_mod_4(A,k2_mod_4(B),k7_mod_4(A,B,C)) = C ) ) ) ).
fof(t35_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_ringcat1(C,A,B)
<=> v1_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t36_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_mod_4(C,A,B)
<=> v1_ringcat1(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t37_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_mod_4(C,A,B)
<=> v3_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t38_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_mod_4(C,A,B)
<=> v2_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t39_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v4_mod_4(C,A,B)
<=> v5_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t40_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_mod_4(C,A,B)
<=> v4_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t41_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v6_mod_4(C,A,B)
<=> v7_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t42_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v7_mod_4(C,A,B)
<=> v6_mod_4(k7_mod_4(A,B,C),A,k2_mod_4(B)) ) ) ) ) ).
fof(t43_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v8_mod_4(B,A)
<=> v1_mod_4(k7_mod_4(A,A,B),A,k2_mod_4(A)) ) ) ) ).
fof(t44_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v9_mod_4(B,A)
<=> v1_ringcat1(k7_mod_4(A,A,B),A,k2_mod_4(A)) ) ) ) ).
fof(t45_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v10_mod_4(B,A)
<=> v7_mod_4(k7_mod_4(A,A,B),A,k2_mod_4(A)) ) ) ) ).
fof(t46_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v11_mod_4(B,A)
<=> v6_mod_4(k7_mod_4(A,A,B),A,k2_mod_4(A)) ) ) ) ).
fof(d18_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_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)) )
=> ( m1_mod_4(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,k2_rlvect_1(A,D,E)) = k2_rlvect_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(d19_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(C,A,B)
=> ( v12_mod_4(C,A,B)
<=> v2_funct_1(C) ) ) ) ) ).
fof(d20_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(C,A,B)
=> ( v13_mod_4(C,A,B)
<=> k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ).
fof(d21_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(C,A,B)
=> ( v14_mod_4(C,A,B)
<=> ( v2_funct_1(C)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(t47_mod_4,axiom,
$true ).
fof(t48_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(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_rlvect_1(A)) = k1_rlvect_1(B)
& k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k5_rlvect_1(A,D)) = k5_rlvect_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,k6_rlvect_1(A,D,E)) = k6_rlvect_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(t49_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,k6_rlvect_1(B,D,E)) = k6_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,D),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E)) ) ) ) ) ) ).
fof(d22_mod_4,axiom,
$true ).
fof(d23_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(D),u1_struct_0(E))
& m2_relset_1(F,u1_struct_0(D),u1_struct_0(E)) )
=> ( m2_mod_4(F,A,B,C,D,E)
<=> ( ! [G] :
( m1_subset_1(G,u1_struct_0(D))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> k8_funct_2(u1_struct_0(D),u1_struct_0(E),F,k4_rlvect_1(D,G,H)) = k4_rlvect_1(E,k8_funct_2(u1_struct_0(D),u1_struct_0(E),F,G),k8_funct_2(u1_struct_0(D),u1_struct_0(E),F,H)) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(D))
=> k8_funct_2(u1_struct_0(D),u1_struct_0(E),F,k6_vectsp_1(A,D,G,H)) = k6_vectsp_1(B,E,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,G),k8_funct_2(u1_struct_0(D),u1_struct_0(E),F,H)) ) ) ) ) ) ) ) ) ) ) ).
fof(t50_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> m2_mod_4(k8_mod_4(D,E),A,B,C,D,E) ) ) ) ) ) ).
fof(d24_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ! [F] :
( m2_mod_4(F,A,B,C,D,E)
=> ( r1_mod_4(A,B,C,D,E,F)
<=> v2_funct_1(F) ) ) ) ) ) ) ) ).
fof(d25_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ! [F] :
( m2_mod_4(F,A,B,C,D,E)
=> ( r2_mod_4(A,B,C,D,E,F)
<=> k2_relat_1(F) = u1_struct_0(E) ) ) ) ) ) ) ) ).
fof(d26_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ! [F] :
( m2_mod_4(F,A,B,C,D,E)
=> ( r3_mod_4(A,B,C,D,E,F)
<=> ( v2_funct_1(F)
& k2_relat_1(F) = u1_struct_0(E) ) ) ) ) ) ) ) ) ).
fof(d27_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_mod_4(D,A,A,B,C,C)
=> ( r4_mod_4(A,B,C,D)
<=> ( v2_funct_1(D)
& k2_relat_1(D) = u1_struct_0(C) ) ) ) ) ) ) ).
fof(t51_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( m2_mod_4(D,A,A,k9_mod_4(A),B,C)
<=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(C),D,k4_rlvect_1(B,E,F)) = k4_rlvect_1(C,k8_funct_2(u1_struct_0(B),u1_struct_0(C),D,E),k8_funct_2(u1_struct_0(B),u1_struct_0(C),D,F)) ) )
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(C),D,k6_vectsp_1(A,B,E,F)) = k6_vectsp_1(A,C,E,k8_funct_2(u1_struct_0(B),u1_struct_0(C),D,F)) ) ) ) ) ) ) ) ) ).
fof(d28_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_mod_4(D,A,A,k9_mod_4(A),B,C)
=> ( v15_mod_4(D,A,B,C)
<=> v2_funct_1(D) ) ) ) ) ) ).
fof(d29_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_mod_4(D,A,A,k9_mod_4(A),B,C)
=> ( v16_mod_4(D,A,B,C)
<=> k2_relat_1(D) = u1_struct_0(C) ) ) ) ) ) ).
fof(d30_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_mod_4(D,A,A,k9_mod_4(A),B,C)
=> ( v17_mod_4(D,A,B,C)
<=> ( v2_funct_1(D)
& k2_relat_1(D) = u1_struct_0(C) ) ) ) ) ) ) ).
fof(d31_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_mod_4(C,A,A,k9_mod_4(A),B,B)
=> ( v18_mod_4(C,A,B)
<=> ( v2_funct_1(C)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(dt_m1_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_mod_4(C,A,B)
=> ( 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)) ) ) ) ).
fof(existence_m1_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ? [C] : m1_mod_4(C,A,B) ) ).
fof(dt_m2_mod_4,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_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))
& ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A)
& ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ! [F] :
( m2_mod_4(F,A,B,C,D,E)
=> ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(D),u1_struct_0(E))
& m2_relset_1(F,u1_struct_0(D),u1_struct_0(E)) ) ) ) ).
fof(existence_m2_mod_4,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_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))
& ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v12_vectsp_1(D,A)
& l4_vectsp_1(D,A)
& ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v12_vectsp_1(E,B)
& l4_vectsp_1(E,B) )
=> ? [F] : m2_mod_4(F,A,B,C,D,E) ) ).
fof(dt_k1_mod_4,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(k1_mod_4(A,B,C,D))
& v1_funct_2(k1_mod_4(A,B,C,D),k2_zfmisc_1(B,A),C)
& m2_relset_1(k1_mod_4(A,B,C,D),k2_zfmisc_1(B,A),C) ) ) ).
fof(redefinition_k1_mod_4,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) )
=> k1_mod_4(A,B,C,D) = k2_funct_4(D) ) ).
fof(dt_k2_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v3_vectsp_1(k2_mod_4(A))
& l3_vectsp_1(k2_mod_4(A)) ) ) ).
fof(dt_k3_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ( v3_vectsp_2(k3_mod_4(A,B),k2_mod_4(A))
& l1_vectsp_2(k3_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(dt_k4_mod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B))
& m1_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B)) )
=> ( v1_funct_1(k4_mod_4(A,B,C))
& v1_funct_2(k4_mod_4(A,B,C),k2_zfmisc_1(u1_struct_0(k3_mod_4(A,B)),u1_struct_0(k2_mod_4(A))),u1_struct_0(k3_mod_4(A,B)))
& m2_relset_1(k4_mod_4(A,B,C),k2_zfmisc_1(u1_struct_0(k3_mod_4(A,B)),u1_struct_0(k2_mod_4(A))),u1_struct_0(k3_mod_4(A,B))) ) ) ).
fof(dt_k5_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l1_vectsp_2(B,A) )
=> ( v11_vectsp_1(k5_mod_4(A,B),k2_mod_4(A))
& l4_vectsp_1(k5_mod_4(A,B),k2_mod_4(A)) ) ) ).
fof(dt_k6_mod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l1_vectsp_2(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B))
& m1_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A)),u1_struct_0(B)) )
=> ( v1_funct_1(k6_mod_4(A,B,C))
& v1_funct_2(k6_mod_4(A,B,C),k2_zfmisc_1(u1_struct_0(k2_mod_4(A)),u1_struct_0(k5_mod_4(A,B))),u1_struct_0(k5_mod_4(A,B)))
& m2_relset_1(k6_mod_4(A,B,C),k2_zfmisc_1(u1_struct_0(k2_mod_4(A)),u1_struct_0(k5_mod_4(A,B))),u1_struct_0(k5_mod_4(A,B))) ) ) ).
fof(dt_k7_mod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_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(k7_mod_4(A,B,C))
& v1_funct_2(k7_mod_4(A,B,C),u1_struct_0(A),u1_struct_0(k2_mod_4(B)))
& m2_relset_1(k7_mod_4(A,B,C),u1_struct_0(A),u1_struct_0(k2_mod_4(B))) ) ) ).
fof(dt_k8_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> m1_mod_4(k8_mod_4(A,B),A,B) ) ).
fof(redefinition_k8_mod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> k8_mod_4(A,B) = k8_grcat_1(A,B) ) ).
fof(dt_k9_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ( v14_mod_4(k9_mod_4(A),A,A)
& m1_mod_4(k9_mod_4(A),A,A) ) ) ).
fof(redefinition_k9_mod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> k9_mod_4(A) = k7_grcat_1(A) ) ).
%------------------------------------------------------------------------------