SET007 Axioms: SET007+759.ax
%------------------------------------------------------------------------------
% File : SET007+759 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Bessel's Inequality
% 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 : bhsp_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 2 unt; 0 def)
% Number of atoms : 515 ( 58 equ)
% Maximal formula atoms : 32 ( 14 avg)
% Number of connectives : 524 ( 45 ~; 3 |; 309 &)
% ( 6 <=>; 161 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 12 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 34 ( 32 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 7 con; 0-6 aty)
% Number of variables : 136 ( 127 !; 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_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ? [B] :
( m1_bhsp_5(B,A)
& v1_finset_1(B) ) ) ).
fof(rc2_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ? [B] :
( m2_bhsp_5(B,A)
& v1_finset_1(B) ) ) ).
fof(t1_bhsp_5,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ( v2_funct_1(B)
& v2_funct_1(C)
& k2_relat_1(B) = k2_relat_1(C) )
=> ( k5_finsop_1(B) = k5_finsop_1(C)
& ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
& v3_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
& m2_relset_1(D,k5_finsop_1(B),k5_finsop_1(B))
& C = k5_relat_1(D,B)
& k1_relat_1(D) = k5_finsop_1(B)
& k2_relat_1(D) = k5_finsop_1(B) ) ) ) ) ) ).
fof(d1_bhsp_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( ( v1_binop_1(B,A)
& v2_binop_1(B,A)
& v1_setwiseo(B,A) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D] :
( m1_subset_1(D,A)
=> ( D = k1_bhsp_5(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,A)
& v2_funct_1(E)
& k2_relat_1(E) = C
& D = k2_finsop_1(A,E,B) ) ) ) ) ) ) ) ).
fof(d2_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( B != k1_xboole_0
=> k2_bhsp_5(A,B) = k1_bhsp_5(u1_struct_0(A),u1_rlvect_1(A),B) )
& ( B = k1_xboole_0
=> k2_bhsp_5(A,B) = k1_rlvect_1(A) ) ) ) ) ).
fof(d3_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k3_bhsp_5(A,B,C,D) = k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(C,D))) ) ) ) ) ).
fof(d4_bhsp_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A) )
=> ! [D] :
( m2_finseq_1(D,B)
=> k4_bhsp_5(A,B,C,D) = k5_relat_1(D,C) ) ) ) ) ).
fof(d5_bhsp_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v1_setwiseo(C,A) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> ( r1_tarski(D,k1_relat_1(E))
=> ! [F] :
( m1_subset_1(F,A)
=> ( F = k5_bhsp_5(A,B,C,D,E)
<=> ? [G] :
( m2_finseq_1(G,B)
& v2_funct_1(G)
& k2_relat_1(G) = D
& F = k2_finsop_1(A,k4_bhsp_5(A,B,E,G),C) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( D = k6_bhsp_5(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,u1_struct_0(A))
& v2_funct_1(E)
& k2_relat_1(E) = C
& ? [F] :
( m2_finseq_1(F,k1_numbers)
& k5_finsop_1(F) = k5_finsop_1(E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k5_finsop_1(F))
=> k1_funct_1(F,G) = k3_bhsp_5(A,B,E,G) ) )
& D = k2_finsop_1(k1_numbers,F,k33_binop_2) ) ) ) ) ) ) ) ).
fof(d7_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( C != k1_xboole_0
=> k7_bhsp_5(A,B,C) = k6_bhsp_5(A,B,C) )
& ( C = k1_xboole_0
=> k7_bhsp_5(A,B,C) = np__0 ) ) ) ) ) ).
fof(d8_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_bhsp_5(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( C = D
| k2_bhsp_1(A,C,D) = np__0 ) ) ) ) ) ) ) ).
fof(t2_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> m1_bhsp_5(k1_xboole_0,A) ) ).
fof(d9_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( m2_bhsp_5(B,A)
<=> ( m1_bhsp_5(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> k2_bhsp_1(A,C,C) = np__1 ) ) ) ) ) ) ).
fof(t3_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> m2_bhsp_5(k1_xboole_0,A) ) ).
fof(t4_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k1_rlvect_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_bhsp_1(A,B,C) = np__0 ) ) ) ) ).
fof(t5_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_binop_2(k7_square_1(k3_bhsp_1(A,k4_rlvect_1(A,B,C))),k7_square_1(k3_bhsp_1(A,k6_rlvect_1(A,B,C)))) = k9_binop_2(k11_binop_2(np__2,k7_square_1(k3_bhsp_1(A,B))),k11_binop_2(np__2,k7_square_1(k3_bhsp_1(A,C)))) ) ) ) ).
fof(t6_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_bhsp_1(A,B,C)
=> k7_square_1(k3_bhsp_1(A,k4_rlvect_1(A,B,C))) = k9_binop_2(k7_square_1(k3_bhsp_1(A,B)),k7_square_1(k3_bhsp_1(A,C))) ) ) ) ) ).
fof(t7_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( ( r1_xreal_0(np__1,k3_finseq_1(B))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k5_finsop_1(B))
& r2_hidden(D,k5_finsop_1(B)) )
=> ( C = D
| k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(B,C),k1_funct_1(B,D))) = np__0 ) ) ) ) )
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ( ( k5_finsop_1(B) = k5_finsop_1(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k5_finsop_1(C))
=> k1_funct_1(C,D) = k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(B,D),k1_funct_1(B,D))) ) ) )
=> k2_bhsp_1(A,k2_finsop_1(u1_struct_0(A),B,u1_rlvect_1(A)),k2_finsop_1(u1_struct_0(A),B,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,C,k33_binop_2) ) ) ) ) ) ).
fof(t8_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( r1_xreal_0(np__1,k3_finseq_1(C))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k5_finsop_1(C) = k5_finsop_1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k5_finsop_1(D))
=> k1_funct_1(D,E) = k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(C,E))) ) ) )
=> k2_bhsp_1(A,B,k2_finsop_1(u1_struct_0(A),C,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,D,k33_binop_2) ) ) ) ) ) ) ).
fof(t9_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( r1_tarski(B,k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> ( D = E
| k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E))) = np__0 ) ) ) ) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,B)
=> k1_funct_1(D,E) = k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(C,E),k1_funct_1(C,E))) ) ) )
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(A))
=> ( ( v2_funct_1(E)
& k2_relat_1(E) = B )
=> k1_funct_1(u1_bhsp_1(A),k4_tarski(k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),C,E),u1_rlvect_1(A)),k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),C,E),u1_rlvect_1(A)))) = k2_finsop_1(k1_numbers,k4_bhsp_5(k1_numbers,u1_struct_0(A),D,E),k33_binop_2) ) ) ) ) ) ) ) ) ).
fof(t10_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> ( r1_tarski(C,k1_relat_1(D))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),k1_numbers)
& m2_relset_1(E,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(C,k1_relat_1(E))
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(F,C)
=> k1_funct_1(E,F) = k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(D,F))) ) ) )
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ( ( v2_funct_1(F)
& k2_relat_1(F) = C )
=> k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),D,F),u1_rlvect_1(A)))) = k2_finsop_1(k1_numbers,k4_bhsp_5(k1_numbers,u1_struct_0(A),E,F),k33_binop_2) ) ) ) ) ) ) ) ) ) ).
fof(t11_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_finset_1(C)
& m2_bhsp_5(C,A) )
=> ( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(C,k1_relat_1(D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,C)
=> k1_funct_1(D,E) = k7_square_1(k2_bhsp_1(A,B,E)) ) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( r1_tarski(C,k1_relat_1(E))
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(F,C)
=> k1_funct_1(E,F) = k3_rlvect_1(A,F,k2_bhsp_1(A,B,F)) ) ) )
=> k2_bhsp_1(A,B,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,E)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,D) ) ) ) ) ) ) ) ) ) ).
fof(t12_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_finset_1(C)
& m2_bhsp_5(C,A) )
=> ( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( r1_tarski(C,k1_relat_1(D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,C)
=> k1_funct_1(D,E) = k3_rlvect_1(A,E,k2_bhsp_1(A,B,E)) ) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),k1_numbers)
& m2_relset_1(E,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(C,k1_relat_1(E))
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(F,C)
=> k1_funct_1(E,F) = k7_square_1(k2_bhsp_1(A,B,F)) ) ) )
=> k2_bhsp_1(A,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,D),k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,D)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,E) ) ) ) ) ) ) ) ) ) ).
fof(t13_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_finset_1(C)
& m2_bhsp_5(C,A) )
=> ( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(C,k1_relat_1(D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,C)
=> k1_funct_1(D,E) = k7_square_1(k2_bhsp_1(A,B,E)) ) ) )
=> r1_xreal_0(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,D),k7_square_1(k3_bhsp_1(A,B))) ) ) ) ) ) ) ) ).
fof(t14_bhsp_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v1_setwiseo(C,A) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(B)) )
=> ( r1_xboole_0(D,E)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,A)
& m2_relset_1(F,B,A) )
=> ( ( r1_tarski(D,k1_relat_1(F))
& r1_tarski(E,k1_relat_1(F)) )
=> ! [G] :
( ( v1_finset_1(G)
& m1_subset_1(G,k1_zfmisc_1(B)) )
=> ( G = k2_xboole_0(D,E)
=> k5_bhsp_5(A,B,C,G,F) = k2_binop_1(A,A,A,C,k5_bhsp_5(A,B,C,D,F),k5_bhsp_5(A,B,C,E,F)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_m1_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_bhsp_5(B,A)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ? [B] : m1_bhsp_5(B,A) ) ).
fof(dt_m2_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_bhsp_5(B,A)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m2_bhsp_5,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ? [B] : m2_bhsp_5(B,A) ) ).
fof(dt_k1_bhsp_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k1_bhsp_5(A,B,C),A) ) ).
fof(dt_k2_bhsp_5,axiom,
$true ).
fof(dt_k3_bhsp_5,axiom,
$true ).
fof(dt_k4_bhsp_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& m1_finseq_1(D,B) )
=> m2_finseq_1(k4_bhsp_5(A,B,C,D),A) ) ).
fof(dt_k5_bhsp_5,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& v1_funct_1(E)
& v1_funct_2(E,B,A)
& m1_relset_1(E,B,A) )
=> m1_subset_1(k5_bhsp_5(A,B,C,D,E),A) ) ).
fof(dt_k6_bhsp_5,axiom,
! [A,B,C] :
( ( ~ 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_bhsp_1(A)
& l1_bhsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_bhsp_5(A,B,C),k1_numbers) ) ).
fof(dt_k7_bhsp_5,axiom,
! [A,B,C] :
( ( ~ 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_bhsp_1(A)
& l1_bhsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k7_bhsp_5(A,B,C),k1_numbers) ) ).
%------------------------------------------------------------------------------