SET007 Axioms: SET007+819.ax
%------------------------------------------------------------------------------
% File : SET007+819 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Exponential Function on Banach Algebra
% 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 : lopban_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 0 unt; 0 def)
% Number of atoms : 1026 ( 77 equ)
% Maximal formula atoms : 23 ( 15 avg)
% Number of connectives : 1031 ( 70 ~; 0 |; 716 &)
% ( 10 <=>; 235 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 13 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-3 aty)
% Number of functors : 51 ( 51 usr; 5 con; 0-4 aty)
% Number of variables : 217 ( 215 !; 2 ?)
% 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_lopban_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k1_lopban_4(A,B))
& v1_funct_1(k1_lopban_4(A,B))
& v1_funct_2(k1_lopban_4(A,B),k5_numbers,u1_struct_0(A))
& v4_normsp_1(k1_lopban_4(A,B),A)
& v1_lopban_3(k1_lopban_4(A,B),A)
& v2_lopban_3(k1_lopban_4(A,B),A) ) ) ).
fof(d1_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_lopban_4(A,B,C)
<=> k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ).
fof(t1_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A)
& k8_normsp_1(A,k4_normsp_1(A,B,C)) = k1_rlvect_1(A) )
=> k8_normsp_1(A,B) = k8_normsp_1(A,C) ) ) ) ) ).
fof(t2_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,B,D) = C )
=> k8_normsp_1(A,B) = C ) ) ) ) ).
fof(t3_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> v4_normsp_1(k8_lopban_3(A,B,C),A) ) ) ) ) ).
fof(t4_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> v4_normsp_1(k6_lopban_3(A,C,B),A) ) ) ) ) ).
fof(t5_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> v4_normsp_1(k7_lopban_3(A,C,B),A) ) ) ) ) ).
fof(t6_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> k8_normsp_1(A,k6_lopban_3(A,C,B)) = k1_group_1(A,B,k8_normsp_1(A,C)) ) ) ) ) ).
fof(t7_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v4_normsp_1(C,A)
=> k8_normsp_1(A,k7_lopban_3(A,C,B)) = k1_group_1(A,k8_normsp_1(A,C),B) ) ) ) ) ).
fof(t8_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v4_normsp_1(B,A)
& v4_normsp_1(C,A) )
=> k8_normsp_1(A,k8_lopban_3(A,B,C)) = k1_group_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) ).
fof(t9_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( k1_lopban_3(A,k6_lopban_3(A,C,B)) = k6_lopban_3(A,k1_lopban_3(A,C),B)
& k1_lopban_3(A,k7_lopban_3(A,C,B)) = k7_lopban_3(A,k1_lopban_3(A,C),B) ) ) ) ) ).
fof(t10_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> r1_xreal_0(k1_normsp_1(A,k2_normsp_1(A,k1_lopban_3(A,C),B)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_normsp_1(A,C)),B)) ) ) ) ).
fof(t11_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,B)
=> k2_normsp_1(A,C,E) = k2_normsp_1(A,D,E) ) )
=> k2_normsp_1(A,k1_lopban_3(A,C),B) = k2_normsp_1(A,k1_lopban_3(A,D),B) ) ) ) ) ) ).
fof(t12_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k1_normsp_1(A,k2_normsp_1(A,B,D)),k2_seq_1(k5_numbers,k1_numbers,C,D)) )
& v4_seq_2(C)
& k2_seq_2(C) = np__0 )
=> ( v4_normsp_1(B,A)
& k8_normsp_1(A,B) = k1_rlvect_1(A) ) ) ) ) ) ).
fof(d2_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k1_lopban_4(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,C,D) = k3_rlvect_1(A,k11_lopban_3(A,B,D),k6_real_1(np__1,k5_sin_cos(D))) ) ) ) ) ) ).
fof(t13_lopban_4,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_real_1(k5_sin_cos(k5_binarith(A,np__1)),A) = k5_sin_cos(A) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> k4_real_1(k5_sin_cos(k5_binarith(A,B)),k5_real_1(k1_nat_1(A,np__1),B)) = k5_sin_cos(k5_binarith(k1_nat_1(A,np__1),B)) ) ) ) ) ).
fof(d3_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k2_lopban_4(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k6_real_1(k5_sin_cos(A),k4_real_1(k5_sin_cos(C),k5_sin_cos(k5_binarith(A,C)))) )
& ( ~ r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 ) ) ) ) ) ) ).
fof(d4_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k3_lopban_4(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k6_real_1(np__1,k4_real_1(k5_sin_cos(C),k5_sin_cos(k5_binarith(A,C)))) )
& ( ~ r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 ) ) ) ) ) ) ).
fof(d5_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k4_lopban_4(A,B)
<=> ( k2_normsp_1(A,C,np__0) = k1_rlvect_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,C,k1_nat_1(D,np__1)) = k2_normsp_1(A,B,D) ) ) ) ) ) ) ).
fof(d6_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( E = k5_lopban_4(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_group_1(B,k3_rlvect_1(B,k11_lopban_3(B,C,F),k2_seq_1(k5_numbers,k1_numbers,k2_lopban_4(A),F)),k11_lopban_3(B,D,k5_binarith(A,F))) )
& ( ~ r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) ) ).
fof(d7_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( E = k6_lopban_4(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_group_1(B,k3_rlvect_1(B,k11_lopban_3(B,C,F),k2_seq_1(k5_numbers,k1_numbers,k3_lopban_4(A),F)),k11_lopban_3(B,D,k5_binarith(A,F))) )
& ( ~ r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) ) ).
fof(d8_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( E = k7_lopban_4(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_group_1(B,k2_normsp_1(B,k1_lopban_4(B,C),F),k2_normsp_1(B,k1_lopban_3(B,k1_lopban_4(B,D)),k5_binarith(A,F))) )
& ( ~ r1_xreal_0(F,A)
=> k2_normsp_1(B,E,F) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) ) ).
fof(d9_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> ( E = k8_lopban_4(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,D)
=> k2_normsp_1(A,E,F) = k1_group_1(A,k2_normsp_1(A,k1_lopban_4(A,B),F),k6_rlvect_1(A,k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,C)),D),k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,C)),k5_binarith(D,F)))) )
& ( ~ r1_xreal_0(F,D)
=> k2_normsp_1(A,E,F) = k1_rlvect_1(A) ) ) ) ) ) ) ) ) ) ).
fof(t14_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k2_normsp_1(A,k1_lopban_4(A,B),k1_nat_1(C,np__1)) = k1_group_1(A,k3_rlvect_1(A,B,k6_real_1(np__1,k1_nat_1(C,np__1))),k2_normsp_1(A,k1_lopban_4(A,B),C))
& k2_normsp_1(A,k1_lopban_4(A,B),np__0) = k2_group_1(A)
& r1_xreal_0(k1_normsp_1(A,k2_normsp_1(A,k1_lopban_4(A,B),C)),k2_seq_1(k5_numbers,k1_numbers,k7_sin_cos(k1_normsp_1(A,B)),C)) ) ) ) ) ).
fof(t15_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ~ r1_xreal_0(B,np__0)
=> k2_normsp_1(A,k4_lopban_4(A,C),B) = k2_normsp_1(A,C,k5_binarith(B,np__1)) ) ) ) ) ).
fof(t16_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k2_normsp_1(A,k1_lopban_3(A,C),B) = k4_rlvect_1(A,k2_normsp_1(A,k1_lopban_3(A,k4_lopban_4(A,C)),B),k2_normsp_1(A,C,B)) ) ) ) ).
fof(t17_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lopban_4(A,C,D)
=> k11_lopban_3(A,k4_rlvect_1(A,C,D),B) = k2_normsp_1(A,k1_lopban_3(A,k5_lopban_4(B,A,C,D)),B) ) ) ) ) ) ).
fof(t18_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k6_lopban_4(D,A,B,C) = k6_normsp_1(A,k5_lopban_4(D,A,B,C),k6_real_1(np__1,k5_sin_cos(D))) ) ) ) ) ).
fof(t19_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lopban_4(A,C,D)
=> k3_rlvect_1(A,k11_lopban_3(A,k4_rlvect_1(A,C,D),B),k6_real_1(np__1,k5_sin_cos(B))) = k2_normsp_1(A,k1_lopban_3(A,k6_lopban_4(B,A,C,D)),B) ) ) ) ) ) ).
fof(t20_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ( v2_lopban_3(k1_lopban_4(A,k1_rlvect_1(A)),A)
& k2_lopban_3(A,k1_lopban_4(A,k1_rlvect_1(A))) = k2_group_1(A) ) ) ).
fof(t21_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_normsp_1(A,k1_lopban_4(A,B),np__0) = k2_group_1(A)
& k2_normsp_1(A,k5_lopban_4(np__0,A,B,C),np__0) = k2_group_1(A) ) ) ) ) ).
fof(t22_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,E)
=> k2_normsp_1(A,k7_lopban_4(k1_nat_1(E,np__1),A,B,C),D) = k4_rlvect_1(A,k2_normsp_1(A,k7_lopban_4(E,A,B,C),D),k2_normsp_1(A,k6_lopban_4(k1_nat_1(E,np__1),A,B,C),D)) ) ) ) ) ) ) ).
fof(t23_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,k1_lopban_3(A,k7_lopban_4(k1_nat_1(D,np__1),A,B,C)),D) = k4_rlvect_1(A,k2_normsp_1(A,k1_lopban_3(A,k7_lopban_4(D,A,B,C)),D),k2_normsp_1(A,k1_lopban_3(A,k6_lopban_4(k1_nat_1(D,np__1),A,B,C)),D)) ) ) ) ) ).
fof(t24_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,k1_lopban_4(A,B),D) = k2_normsp_1(A,k6_lopban_4(D,A,B,C),D) ) ) ) ) ).
fof(t25_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lopban_4(A,C,D)
=> k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,k4_rlvect_1(A,C,D))),B) = k2_normsp_1(A,k1_lopban_3(A,k7_lopban_4(B,A,C,D)),B) ) ) ) ) ) ).
fof(t26_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lopban_4(A,C,D)
=> k6_rlvect_1(A,k1_group_1(A,k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,C)),B),k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,D)),B)),k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,k4_rlvect_1(A,C,D))),B)) = k2_normsp_1(A,k1_lopban_3(A,k8_lopban_4(A,C,D,B)),B) ) ) ) ) ) ).
fof(t27_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k7_sin_cos(k1_normsp_1(A,B)),C)) ) ) ) ).
fof(t28_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_normsp_1(A,k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,B)),C)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C),k2_series_1(k7_sin_cos(k1_normsp_1(A,B))))
& r1_xreal_0(k1_normsp_1(A,k2_normsp_1(A,k1_lopban_3(A,k1_lopban_4(A,B)),C)),k2_series_1(k7_sin_cos(k1_normsp_1(A,B)))) ) ) ) ) ).
fof(t29_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_xreal_0(np__1,k2_series_1(k7_sin_cos(k1_normsp_1(A,B)))) ) ) ).
fof(t30_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C)
& ( r1_xreal_0(C,D)
=> k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C))) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k1_normsp_1(A,B))),C)) ) ) ) ) ) ) ).
fof(t31_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_normsp_1(A,k8_lopban_4(A,B,C,D))),E)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_normsp_1(A,k8_lopban_4(A,B,C,D))),E) ) ) ) ) ) ).
fof(t32_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_normsp_1(A,k8_lopban_4(A,B,C,F))),F))) ) ) ) ) ) ) ) ).
fof(t33_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k2_normsp_1(A,k1_lopban_3(A,k8_lopban_4(A,B,C,E)),E) )
=> ( v4_normsp_1(D,A)
& k8_normsp_1(A,D) = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(d10_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(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)) )
=> ( B = k9_lopban_4(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k2_lopban_3(A,k1_lopban_4(A,C)) ) ) ) ) ).
fof(d11_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k10_lopban_4(A,B) = k8_funct_2(u1_struct_0(A),u1_struct_0(A),k9_lopban_4(A),B) ) ) ).
fof(t34_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k10_lopban_4(A,B) = k2_lopban_3(A,k1_lopban_4(A,B)) ) ) ).
fof(t35_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_lopban_4(A,B,C)
=> ( k10_lopban_4(A,k4_rlvect_1(A,B,C)) = k1_group_1(A,k10_lopban_4(A,B),k10_lopban_4(A,C))
& k10_lopban_4(A,k4_rlvect_1(A,C,B)) = k1_group_1(A,k10_lopban_4(A,C),k10_lopban_4(A,B))
& k10_lopban_4(A,k4_rlvect_1(A,B,C)) = k10_lopban_4(A,k4_rlvect_1(A,C,B))
& r1_lopban_4(A,k10_lopban_4(A,B),k10_lopban_4(A,C)) ) ) ) ) ) ).
fof(t36_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_lopban_4(A,B,C)
=> k1_group_1(A,B,k10_lopban_4(A,C)) = k1_group_1(A,k10_lopban_4(A,C),B) ) ) ) ) ).
fof(t37_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> k10_lopban_4(A,k1_rlvect_1(A)) = k2_group_1(A) ) ).
fof(t38_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_group_1(A,k10_lopban_4(A,B),k10_lopban_4(A,k5_rlvect_1(A,B))) = k2_group_1(A)
& k1_group_1(A,k10_lopban_4(A,k5_rlvect_1(A,B)),k10_lopban_4(A,B)) = k2_group_1(A) ) ) ) ).
fof(t39_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v4_lopban_3(k10_lopban_4(A,B),A)
& k9_lopban_3(A,k10_lopban_4(A,B)) = k10_lopban_4(A,k5_rlvect_1(A,B))
& v4_lopban_3(k10_lopban_4(A,k5_rlvect_1(A,B)),A)
& k9_lopban_3(A,k10_lopban_4(A,k5_rlvect_1(A,B))) = k10_lopban_4(A,B) ) ) ) ).
fof(t40_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> r1_lopban_4(A,k3_rlvect_1(A,B,C),k3_rlvect_1(A,B,D)) ) ) ) ) ).
fof(t41_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( k1_group_1(A,k10_lopban_4(A,k3_rlvect_1(A,B,C)),k10_lopban_4(A,k3_rlvect_1(A,B,D))) = k10_lopban_4(A,k3_rlvect_1(A,B,k3_real_1(C,D)))
& k1_group_1(A,k10_lopban_4(A,k3_rlvect_1(A,B,D)),k10_lopban_4(A,k3_rlvect_1(A,B,C))) = k10_lopban_4(A,k3_rlvect_1(A,B,k3_real_1(D,C)))
& k10_lopban_4(A,k3_rlvect_1(A,B,k3_real_1(C,D))) = k10_lopban_4(A,k3_rlvect_1(A,B,k3_real_1(D,C)))
& r1_lopban_4(A,k10_lopban_4(A,k3_rlvect_1(A,B,C)),k10_lopban_4(A,k3_rlvect_1(A,B,D))) ) ) ) ) ) ).
fof(s1_lopban_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(f1_s1_lopban_4))
& m2_relset_1(B,k5_numbers,u1_struct_0(f1_s1_lopban_4))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k2_normsp_1(f1_s1_lopban_4,B,C) = f2_s1_lopban_4(A,C) )
& ( ~ r1_xreal_0(C,A)
=> k2_normsp_1(f1_s1_lopban_4,B,C) = k1_rlvect_1(f1_s1_lopban_4) ) ) ) ) ) ).
fof(symmetry_r1_lopban_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_lopban_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r1_lopban_4(A,B,C)
=> r1_lopban_4(A,C,B) ) ) ).
fof(dt_k1_lopban_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k1_lopban_4(A,B))
& v1_funct_2(k1_lopban_4(A,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k1_lopban_4(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k2_lopban_4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k2_lopban_4(A))
& v1_funct_2(k2_lopban_4(A),k5_numbers,k1_numbers)
& m2_relset_1(k2_lopban_4(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_lopban_4,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k3_lopban_4(A))
& v1_funct_2(k3_lopban_4(A),k5_numbers,k1_numbers)
& m2_relset_1(k3_lopban_4(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k4_lopban_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k4_lopban_4(A,B))
& v1_funct_2(k4_lopban_4(A,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k4_lopban_4(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k5_lopban_4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_funct_1(k5_lopban_4(A,B,C,D))
& v1_funct_2(k5_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B))
& m2_relset_1(k5_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B)) ) ) ).
fof(dt_k6_lopban_4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_funct_1(k6_lopban_4(A,B,C,D))
& v1_funct_2(k6_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B))
& m2_relset_1(k6_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B)) ) ) ).
fof(dt_k7_lopban_4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& v4_group_1(B)
& v2_funcsdom(B)
& v5_lopban_2(B)
& l1_lopban_2(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_funct_1(k7_lopban_4(A,B,C,D))
& v1_funct_2(k7_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B))
& m2_relset_1(k7_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(B)) ) ) ).
fof(dt_k8_lopban_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k5_numbers) )
=> ( v1_funct_1(k8_lopban_4(A,B,C,D))
& v1_funct_2(k8_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(A))
& m2_relset_1(k8_lopban_4(A,B,C,D),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k9_lopban_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A) )
=> ( v1_funct_1(k9_lopban_4(A))
& v1_funct_2(k9_lopban_4(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k9_lopban_4(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k10_lopban_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_group_1(A)
& v2_funcsdom(A)
& v5_lopban_2(A)
& l1_lopban_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k10_lopban_4(A,B),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------