SET007 Axioms: SET007+805.ax
%------------------------------------------------------------------------------
% File : SET007+805 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Witt's Proof of the Wedderburn Theorem
% 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 : weddwitt [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 67 ( 0 unt; 0 def)
% Number of atoms : 937 ( 67 equ)
% Maximal formula atoms : 32 ( 13 avg)
% Number of connectives : 1001 ( 131 ~; 3 |; 703 &)
% ( 17 <=>; 147 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 13 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 52 ( 51 usr; 0 prp; 1-3 aty)
% Number of functors : 58 ( 58 usr; 4 con; 0-4 aty)
% Number of variables : 156 ( 147 !; 9 ?)
% 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_weddwitt,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(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) ) ).
fof(fc1_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k10_group_5(A))
& v1_group_1(k10_group_5(A))
& v2_group_1(k10_group_5(A))
& v3_group_1(k10_group_5(A))
& v4_group_1(k10_group_5(A))
& v6_group_1(k10_group_5(A))
& v11_monoid_0(k10_group_5(A))
& v12_monoid_0(k10_group_5(A))
& v13_monoid_0(k10_group_5(A))
& v14_monoid_0(k10_group_5(A))
& v15_monoid_0(k10_group_5(A))
& v16_monoid_0(k10_group_5(A)) ) ) ).
fof(fc2_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k1_weddwitt(A,B))
& v1_group_1(k1_weddwitt(A,B))
& v2_group_1(k1_weddwitt(A,B))
& v3_group_1(k1_weddwitt(A,B))
& v4_group_1(k1_weddwitt(A,B))
& v6_group_1(k1_weddwitt(A,B))
& v11_monoid_0(k1_weddwitt(A,B))
& v12_monoid_0(k1_weddwitt(A,B))
& v13_monoid_0(k1_weddwitt(A,B))
& v14_monoid_0(k1_weddwitt(A,B))
& v15_monoid_0(k1_weddwitt(A,B))
& v16_monoid_0(k1_weddwitt(A,B)) ) ) ).
fof(fc3_weddwitt,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)) )
=> ~ v1_xboole_0(k7_group_3(A,B)) ) ).
fof(fc4_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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(k4_weddwitt(A))
& v2_group_1(k4_weddwitt(A))
& v4_group_1(k4_weddwitt(A))
& v6_group_1(k4_weddwitt(A))
& v7_group_1(k4_weddwitt(A))
& v3_rlvect_1(k4_weddwitt(A))
& v4_rlvect_1(k4_weddwitt(A))
& v5_rlvect_1(k4_weddwitt(A))
& v6_rlvect_1(k4_weddwitt(A))
& v3_vectsp_1(k4_weddwitt(A))
& v4_vectsp_1(k4_weddwitt(A))
& v5_vectsp_1(k4_weddwitt(A))
& v6_vectsp_1(k4_weddwitt(A))
& v7_vectsp_1(k4_weddwitt(A))
& v8_vectsp_1(k4_weddwitt(A))
& v9_vectsp_1(k4_weddwitt(A))
& ~ v10_vectsp_1(k4_weddwitt(A))
& ~ v3_realset2(k4_weddwitt(A)) ) ) ).
fof(fc5_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k5_weddwitt(A,B))
& v2_group_1(k5_weddwitt(A,B))
& v4_group_1(k5_weddwitt(A,B))
& v6_group_1(k5_weddwitt(A,B))
& v3_rlvect_1(k5_weddwitt(A,B))
& v4_rlvect_1(k5_weddwitt(A,B))
& v5_rlvect_1(k5_weddwitt(A,B))
& v6_rlvect_1(k5_weddwitt(A,B))
& v3_vectsp_1(k5_weddwitt(A,B))
& v4_vectsp_1(k5_weddwitt(A,B))
& v5_vectsp_1(k5_weddwitt(A,B))
& v6_vectsp_1(k5_weddwitt(A,B))
& v7_vectsp_1(k5_weddwitt(A,B))
& v8_vectsp_1(k5_weddwitt(A,B))
& v9_vectsp_1(k5_weddwitt(A,B))
& ~ v10_vectsp_1(k5_weddwitt(A,B))
& ~ v3_realset2(k5_weddwitt(A,B)) ) ) ).
fof(t1_weddwitt,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_prepower(B,A) = np__1
=> ( r1_xreal_0(B,np__1)
| A = np__0 ) ) ) ) ).
fof(t2_weddwitt,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__1)
& ~ r1_xreal_0(B,np__0)
& k3_prepower(D,k1_nat_1(k2_nat_1(A,B),C)) != k4_real_1(k3_prepower(D,A),k3_prepower(D,k1_nat_1(k2_nat_1(A,k5_binarith(B,np__1)),C))) ) ) ) ) ) ).
fof(t3_weddwitt,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(k5_binarith(k3_euler_2(A,B),np__1),k5_binarith(k3_euler_2(A,C),np__1))
=> ( r1_xreal_0(B,np__0)
| r1_xreal_0(A,np__1)
| r1_nat_1(B,C) ) ) ) ) ) ).
fof(t4_weddwitt,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k1_card_1(k1_funct_2(A,B)) = k3_euler_2(B,A) ) ) ) ).
fof(t5_weddwitt,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> r1_nat_1(B,k4_finseq_4(k5_numbers,k5_numbers,A,C)) ) )
=> r1_nat_1(B,k9_wsierp_1(A)) ) ) ) ).
fof(t6_weddwitt,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( ( v2_funct_1(C)
& k2_relat_1(C) = B )
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> k3_wsierp_1(D,E) = k1_card_1(k1_funct_1(C,E)) ) ) )
=> k4_card_1(A) = k9_wsierp_1(D) ) ) ) ) ) ) ).
fof(t7_weddwitt,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] :
( r1_rlvect_1(k1_weddwitt(A,B),C)
=> r1_rlvect_1(A,C) ) ) ) ).
fof(t8_weddwitt,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))
=> ( k1_group_1(A,B,C) = k1_group_1(A,C,B)
<=> m1_subset_1(C,u1_struct_0(k1_weddwitt(A,B))) ) ) ) ) ).
fof(d2_weddwitt,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k7_group_3(A,B))
& m2_relset_1(C,u1_struct_0(A),k7_group_3(A,B)) )
=> ( C = k2_weddwitt(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k7_group_3(A,B),C,D) = k2_group_3(A,B,D) ) ) ) ) ) ).
fof(t9_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_struct_0(C,A,k7_group_3(A,B))
=> k4_card_1(k3_funct_2(u1_struct_0(A),k7_group_3(A,B),k2_weddwitt(A,B),k18_group_2(k7_group_3(A,B),C))) = k9_group_1(k1_weddwitt(A,B)) ) ) ) ).
fof(t10_weddwitt,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_struct_0(C,A,k7_group_3(A,B))
=> ! [D] :
( m1_struct_0(D,A,k7_group_3(A,B))
=> ( C != D
=> r1_xboole_0(k3_funct_2(u1_struct_0(A),k7_group_3(A,B),k2_weddwitt(A,B),k18_group_2(k7_group_3(A,B),C)),k3_funct_2(u1_struct_0(A),k7_group_3(A,B),k2_weddwitt(A,B),k18_group_2(k7_group_3(A,B),D))) ) ) ) ) ) ).
fof(t13_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k9_group_1(A) = k2_nat_1(k4_card_1(k7_group_3(A,B)),k9_group_1(k1_weddwitt(A,B))) ) ) ).
fof(t14_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( r2_hidden(B,k3_weddwitt(A))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k7_group_3(A,C) = B ) ) ) ).
fof(t15_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,k3_weddwitt(A))
=> ( ( v2_funct_1(B)
& k2_relat_1(B) = k3_weddwitt(A) )
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( ( k3_finseq_1(C) = k3_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> k3_wsierp_1(C,D) = k1_card_1(k1_funct_1(B,D)) ) ) )
=> k9_group_1(A) = k9_wsierp_1(C) ) ) ) ) ) ).
fof(t16_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( v2_matrlin(B,A)
& C = k1_vectsp_9(A,B)
& D = k1_card_1(u1_struct_0(A)) )
=> k1_card_1(u1_struct_0(B)) = k3_euler_2(D,C) ) ) ) ) ) ).
fof(t17_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> r1_tarski(u1_struct_0(k4_weddwitt(A)),u1_struct_0(A)) ) ).
fof(t18_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_rlvect_1(k4_weddwitt(A),B)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ).
fof(t19_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> r1_rlvect_1(k4_weddwitt(A),k1_rlvect_1(A)) ) ).
fof(t20_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> r1_rlvect_1(k4_weddwitt(A),k2_group_1(A)) ) ).
fof(t21_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ~ r1_xreal_0(k4_card_1(u1_struct_0(k4_weddwitt(A))),np__1) ) ).
fof(t22_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ( k4_card_1(u1_struct_0(k4_weddwitt(A))) = k4_card_1(u1_struct_0(A))
<=> v7_group_1(A) ) ) ).
fof(t23_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> u1_struct_0(k4_weddwitt(A)) = k2_xboole_0(u1_struct_0(k10_group_5(k1_uniroots(A))),k18_group_2(u1_struct_0(A),k1_rlvect_1(A))) ) ).
fof(t24_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(u1_struct_0(k5_weddwitt(A,B)),u1_struct_0(A)) ) ) ).
fof(t25_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,u1_struct_0(k5_weddwitt(A,B)))
<=> k1_group_1(A,C,B) = k1_group_1(A,B,C) ) ) ) ) ).
fof(t26_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(u1_struct_0(k4_weddwitt(A)),u1_struct_0(k5_weddwitt(A,B))) ) ) ).
fof(t27_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [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))
=> ( ( r2_hidden(C,u1_struct_0(k4_weddwitt(A)))
& r2_hidden(D,u1_struct_0(k5_weddwitt(A,B))) )
=> r2_hidden(k1_group_1(A,C,D),u1_struct_0(k5_weddwitt(A,B))) ) ) ) ) ) ).
fof(t28_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( m1_subset_1(k1_rlvect_1(A),u1_struct_0(k5_weddwitt(A,B)))
& m1_subset_1(k2_group_1(A),u1_struct_0(k5_weddwitt(A,B))) ) ) ) ).
fof(t29_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ r1_xreal_0(k4_card_1(u1_struct_0(k5_weddwitt(A,B))),np__1) ) ) ).
fof(t30_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_uniroots(A)))
=> ( C = B
=> u1_struct_0(k5_weddwitt(A,B)) = k2_xboole_0(u1_struct_0(k1_weddwitt(k1_uniroots(A),C)),k18_group_2(u1_struct_0(A),k1_rlvect_1(A))) ) ) ) ) ).
fof(t31_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_uniroots(A)))
=> ( C = B
=> k9_group_1(k1_weddwitt(k1_uniroots(A),C)) = k5_real_1(k4_card_1(u1_struct_0(k5_weddwitt(A,B))),np__1) ) ) ) ) ).
fof(d6_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,k4_weddwitt(A))
& v12_vectsp_1(B,k4_weddwitt(A))
& l4_vectsp_1(B,k4_weddwitt(A)) )
=> ( B = k6_weddwitt(A)
<=> ( g1_rlvect_1(u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B)) = g1_rlvect_1(u1_struct_0(A),u1_rlvect_1(A),u2_struct_0(A))
& u2_vectsp_1(k4_weddwitt(A),B) = k7_relat_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(k4_weddwitt(A)),u1_struct_0(A))) ) ) ) ) ).
fof(t32_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> k4_card_1(u1_struct_0(A)) = k3_euler_2(k4_card_1(u1_struct_0(k4_weddwitt(A))),k1_vectsp_9(k4_weddwitt(A),k6_weddwitt(A))) ) ).
fof(t33_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ~ r1_xreal_0(k1_vectsp_9(k4_weddwitt(A),k6_weddwitt(A)),np__0) ) ).
fof(d7_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,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)
& v11_vectsp_1(C,k4_weddwitt(A))
& v12_vectsp_1(C,k4_weddwitt(A))
& l4_vectsp_1(C,k4_weddwitt(A)) )
=> ( C = k7_weddwitt(A,B)
<=> ( g1_rlvect_1(u1_struct_0(C),u1_rlvect_1(C),u2_struct_0(C)) = g1_rlvect_1(u1_struct_0(k5_weddwitt(A,B)),u1_rlvect_1(k5_weddwitt(A,B)),u2_struct_0(k5_weddwitt(A,B)))
& u2_vectsp_1(k4_weddwitt(A),C) = k7_relat_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(k4_weddwitt(A)),u1_struct_0(k5_weddwitt(A,B)))) ) ) ) ) ) ).
fof(t34_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_card_1(u1_struct_0(k5_weddwitt(A,B))) = k3_euler_2(k4_card_1(u1_struct_0(k4_weddwitt(A))),k1_vectsp_9(k4_weddwitt(A),k7_weddwitt(A,B))) ) ) ).
fof(t35_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ r1_xreal_0(k1_vectsp_9(k4_weddwitt(A),k7_weddwitt(A,B)),np__0) ) ) ).
fof(t36_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( m1_subset_1(B,u1_struct_0(k1_uniroots(A)))
=> r2_int_1(k5_real_1(k3_euler_2(k4_card_1(u1_struct_0(k4_weddwitt(A))),k1_vectsp_9(k4_weddwitt(A),k7_weddwitt(A,B))),np__1),k5_real_1(k3_euler_2(k4_card_1(u1_struct_0(k4_weddwitt(A))),k1_vectsp_9(k4_weddwitt(A),k6_weddwitt(A))),np__1)) ) ) ) ).
fof(t37_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( m1_subset_1(B,u1_struct_0(k1_uniroots(A)))
=> r1_nat_1(k1_vectsp_9(k4_weddwitt(A),k7_weddwitt(A,B)),k1_vectsp_9(k4_weddwitt(A),k6_weddwitt(A))) ) ) ) ).
fof(t38_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> k4_card_1(u1_struct_0(k10_group_5(k1_uniroots(A)))) = k5_real_1(k4_card_1(u1_struct_0(k4_weddwitt(A))),np__1) ) ).
fof(t39_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v6_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> v7_group_1(A) ) ).
fof(t40_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> k2_group_1(k4_weddwitt(A)) = k2_group_1(A) ) ).
fof(t41_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_group_1(k5_weddwitt(A,B)) = k2_group_1(A) ) ) ).
fof(dt_k1_weddwitt,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)) )
=> ( v1_group_1(k1_weddwitt(A,B))
& m1_group_2(k1_weddwitt(A,B),A) ) ) ).
fof(dt_k2_weddwitt,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)) )
=> ( v1_funct_1(k2_weddwitt(A,B))
& v1_funct_2(k2_weddwitt(A,B),u1_struct_0(A),k7_group_3(A,B))
& m2_relset_1(k2_weddwitt(A,B),u1_struct_0(A),k7_group_3(A,B)) ) ) ).
fof(dt_k3_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m1_eqrel_1(k3_weddwitt(A),u1_struct_0(A)) ) ).
fof(dt_k4_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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(k4_weddwitt(A))
& v4_group_1(k4_weddwitt(A))
& v7_group_1(k4_weddwitt(A))
& v3_rlvect_1(k4_weddwitt(A))
& v4_rlvect_1(k4_weddwitt(A))
& v5_rlvect_1(k4_weddwitt(A))
& v6_rlvect_1(k4_weddwitt(A))
& v3_vectsp_1(k4_weddwitt(A))
& v7_vectsp_1(k4_weddwitt(A))
& v8_vectsp_1(k4_weddwitt(A))
& v9_vectsp_1(k4_weddwitt(A))
& ~ v10_vectsp_1(k4_weddwitt(A))
& l3_vectsp_1(k4_weddwitt(A)) ) ) ).
fof(dt_k5_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k5_weddwitt(A,B))
& v4_group_1(k5_weddwitt(A,B))
& v3_rlvect_1(k5_weddwitt(A,B))
& v4_rlvect_1(k5_weddwitt(A,B))
& v5_rlvect_1(k5_weddwitt(A,B))
& v6_rlvect_1(k5_weddwitt(A,B))
& v3_vectsp_1(k5_weddwitt(A,B))
& v6_vectsp_1(k5_weddwitt(A,B))
& v7_vectsp_1(k5_weddwitt(A,B))
& v8_vectsp_1(k5_weddwitt(A,B))
& v9_vectsp_1(k5_weddwitt(A,B))
& ~ v10_vectsp_1(k5_weddwitt(A,B))
& l3_vectsp_1(k5_weddwitt(A,B)) ) ) ).
fof(dt_k6_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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(k6_weddwitt(A))
& v3_rlvect_1(k6_weddwitt(A))
& v4_rlvect_1(k6_weddwitt(A))
& v5_rlvect_1(k6_weddwitt(A))
& v6_rlvect_1(k6_weddwitt(A))
& v11_vectsp_1(k6_weddwitt(A),k4_weddwitt(A))
& v12_vectsp_1(k6_weddwitt(A),k4_weddwitt(A))
& l4_vectsp_1(k6_weddwitt(A),k4_weddwitt(A)) ) ) ).
fof(dt_k7_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k7_weddwitt(A,B))
& v3_rlvect_1(k7_weddwitt(A,B))
& v4_rlvect_1(k7_weddwitt(A,B))
& v5_rlvect_1(k7_weddwitt(A,B))
& v6_rlvect_1(k7_weddwitt(A,B))
& v11_vectsp_1(k7_weddwitt(A,B),k4_weddwitt(A))
& v12_vectsp_1(k7_weddwitt(A,B),k4_weddwitt(A))
& l4_vectsp_1(k7_weddwitt(A,B),k4_weddwitt(A)) ) ) ).
fof(d1_weddwitt,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] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( C = k1_weddwitt(A,B)
<=> u1_struct_0(C) = a_2_0_weddwitt(A,B) ) ) ) ) ).
fof(t11_weddwitt,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))
=> m1_eqrel_1(a_2_1_weddwitt(A,B),u1_struct_0(A)) ) ) ).
fof(t12_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_card_1(a_2_2_weddwitt(A,B)) = k4_card_1(k7_group_3(A,B)) ) ) ).
fof(d3_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k3_weddwitt(A) = a_1_0_weddwitt(A) ) ).
fof(d4_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( B = k4_weddwitt(A)
<=> ( u1_struct_0(B) = a_1_1_weddwitt(A)
& u1_rlvect_1(B) = k1_realset1(u1_rlvect_1(A),u1_struct_0(B))
& u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B))
& u2_struct_0(B) = u2_struct_0(A)
& u1_vectsp_1(B) = k2_group_1(A) ) ) ) ) ).
fof(d5_weddwitt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v3_vectsp_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ( C = k5_weddwitt(A,B)
<=> ( u1_struct_0(C) = a_2_3_weddwitt(A,B)
& u1_rlvect_1(C) = k1_realset1(u1_rlvect_1(A),u1_struct_0(C))
& u1_group_1(C) = k1_realset1(u1_group_1(A),u1_struct_0(C))
& u2_struct_0(C) = u2_struct_0(A)
& u1_vectsp_1(C) = k2_group_1(A) ) ) ) ) ) ).
fof(fraenkel_a_2_0_weddwitt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_weddwitt(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& k1_group_1(B,C,D) = k1_group_1(B,D,C) ) ) ) ).
fof(fraenkel_a_2_1_weddwitt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_1_weddwitt(B,C))
<=> ? [D] :
( m1_struct_0(D,B,k7_group_3(B,C))
& A = k3_funct_2(u1_struct_0(B),k7_group_3(B,C),k2_weddwitt(B,C),k18_group_2(k7_group_3(B,C),D)) ) ) ) ).
fof(fraenkel_a_2_2_weddwitt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v6_group_1(B)
& l1_group_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_2_weddwitt(B,C))
<=> ? [D] :
( m1_struct_0(D,B,k7_group_3(B,C))
& A = k3_funct_2(u1_struct_0(B),k7_group_3(B,C),k2_weddwitt(B,C),k18_group_2(k7_group_3(B,C),D)) ) ) ) ).
fof(fraenkel_a_1_0_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( r2_hidden(A,a_1_0_weddwitt(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& C = k7_group_3(B,D) ) ) ) ) ).
fof(fraenkel_a_1_1_weddwitt,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( r2_hidden(A,a_1_1_weddwitt(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = C
& ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k1_group_1(B,C,D) = k1_group_1(B,D,C) ) ) ) ) ).
fof(fraenkel_a_2_3_weddwitt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_3_weddwitt(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& k1_group_1(B,D,C) = k1_group_1(B,C,D) ) ) ) ).
%------------------------------------------------------------------------------