SET007 Axioms: SET007+746.ax
%------------------------------------------------------------------------------
% File : SET007+746 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Convex Sets and Convex Combinations
% 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 : convex1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 0 unt; 0 def)
% Number of atoms : 513 ( 59 equ)
% Maximal formula atoms : 21 ( 9 avg)
% Number of connectives : 537 ( 79 ~; 10 |; 254 &)
% ( 11 <=>; 183 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-3 aty)
% Number of functors : 45 ( 45 usr; 5 con; 0-4 aty)
% Number of variables : 188 ( 179 !; 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_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_convex1(B,A) ) ) ).
fof(rc2_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v1_convex1(B,A) ) ) ).
fof(d2_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_convex1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r1_xreal_0(E,np__0)
| r1_xreal_0(np__1,E)
| r2_hidden(k2_rlvect_1(A,k3_rlvect_1(A,C,E),k3_rlvect_1(A,D,k5_real_1(np__1,E))),B) ) ) ) ) ) ) ) ) ).
fof(t1_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( v1_convex1(B,A)
=> v1_convex1(k1_convex1(A,B,C),A) ) ) ) ) ).
fof(t2_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_convex1(B,A)
& v1_convex1(C,A) )
=> v1_convex1(k7_rusub_4(A,B,C),A) ) ) ) ) ).
fof(t3_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_convex1(B,A)
& v1_convex1(C,A) )
=> v1_convex1(k1_rusub_5(A,B,C),A) ) ) ) ) ).
fof(t4_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_convex1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ~ r1_tarski(k6_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,k5_real_1(np__1,C))),B) ) ) ) ) ) ).
fof(t5_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_convex1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ~ r1_tarski(k7_rusub_4(A,k1_convex1(A,B,k5_real_1(np__1,C)),k1_convex1(A,B,C)),B) ) ) ) ) ) ).
fof(t6_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_convex1(B,A)
& v1_convex1(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> v1_convex1(k7_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,k5_real_1(np__1,D))),A) ) ) ) ) ) ).
fof(t7_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v1_convex1(B,A)
<=> v1_convex1(k5_rusub_4(A,B,C),A) ) ) ) ) ).
fof(t8_convex1,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)
& l2_rlvect_1(A) )
=> v1_convex1(k3_rusub_4(A,k1_rlsub_1(A)),A) ) ).
fof(t9_convex1,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)
& l2_rlvect_1(A) )
=> v1_convex1(k3_rusub_4(A,k2_rlsub_1(A)),A) ) ).
fof(t10_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k1_xboole_0
=> v1_convex1(B,A) ) ) ) ).
fof(t11_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( v1_convex1(B,A)
& v1_convex1(C,A) )
=> v1_convex1(k7_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,E)),A) ) ) ) ) ) ) ).
fof(t12_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> r1_tarski(k1_convex1(A,B,k3_real_1(C,D)),k6_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,D))) ) ) ) ) ).
fof(t13_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(np__0,C)
& r1_xreal_0(np__0,D)
& v1_convex1(B,A) )
=> k7_rusub_4(A,k1_convex1(A,B,C),k1_convex1(A,B,D)) = k1_convex1(A,B,k3_real_1(C,D)) ) ) ) ) ) ).
fof(t14_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( v1_convex1(B,A)
& v1_convex1(C,A)
& v1_convex1(D,A) )
=> v1_convex1(k7_rusub_4(A,k7_rusub_4(A,k1_convex1(A,B,E),k1_convex1(A,C,F)),k1_convex1(A,D,G)),A) ) ) ) ) ) ) ) ) ).
fof(t15_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v1_convex1(C,A) ) )
=> v1_convex1(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t16_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_rusub_4(B,A)
=> v1_convex1(B,A) ) ) ) ).
fof(d3_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlvect_2(B,A)
=> ( v2_convex1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v2_funct_1(C)
& k2_relat_1(C) = k10_rlvect_2(A,B)
& ? [D] :
( m2_finseq_1(D,k1_numbers)
& k3_finseq_1(D) = k3_finseq_1(C)
& k15_rvsum_1(D) = np__1
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> ( k1_funct_1(D,E) = k1_funct_1(B,k1_funct_1(C,E))
& r1_xreal_0(np__0,k1_funct_1(D,E)) ) ) ) ) ) ) ) ) ).
fof(t21_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlvect_2(B,A)
=> ~ ( v2_convex1(B,A)
& k10_rlvect_2(A,B) = k1_xboole_0 ) ) ) ).
fof(t22_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlvect_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( v2_convex1(B,A)
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,B,C),np__0)
& r2_hidden(C,k10_rlvect_2(A,B)) ) ) ) ) ).
fof(t23_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlvect_2(B,A)
=> ~ ( v2_convex1(B,A)
& B = k11_rlvect_2(A) ) ) ) ).
fof(t24_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_rlvect_2(C,A,k7_rlvect_2(A,B))
=> ( v2_convex1(C,A)
=> ( k8_funct_2(u1_struct_0(A),k1_numbers,C,B) = np__1
& k13_rlvect_2(A,C) = k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,C,B)) ) ) ) ) ) ).
fof(t25_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_rlvect_2(D,A,k8_rlvect_2(A,B,C))
=> ( v2_convex1(D,A)
=> ( B = C
| ( k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,D,B),k8_funct_2(u1_struct_0(A),k1_numbers,D,C)) = np__1
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,B))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))
& k13_rlvect_2(A,D) = k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,D,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,D,C))) ) ) ) ) ) ) ) ).
fof(t26_convex1,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)
& l2_rlvect_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))
=> ! [E] :
( m2_rlvect_2(E,A,k9_rlvect_2(A,B,C,D))
=> ( v2_convex1(E,A)
=> ( B = C
| C = D
| D = B
| ( k3_real_1(k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,E,B),k8_funct_2(u1_struct_0(A),k1_numbers,E,C)),k8_funct_2(u1_struct_0(A),k1_numbers,E,D)) = np__1
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,B))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))
& k13_rlvect_2(A,E) = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,E,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))),k3_rlvect_1(A,D,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))) ) ) ) ) ) ) ) ) ).
fof(t27_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_rlvect_2(C,A)
=> ( ( v2_convex1(C,A)
& k10_rlvect_2(A,C) = k7_rlvect_2(A,B) )
=> k8_funct_2(u1_struct_0(A),k1_numbers,C,B) = np__1 ) ) ) ) ).
fof(t28_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_rlvect_2(D,A)
=> ( ( v2_convex1(D,A)
& k10_rlvect_2(A,D) = k8_rlvect_2(A,B,C) )
=> ( B = C
| ( k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,D,B),k8_funct_2(u1_struct_0(A),k1_numbers,D,C)) = np__1
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,B))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,C)) ) ) ) ) ) ) ) ).
fof(t29_convex1,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)
& l2_rlvect_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))
=> ! [E] :
( m1_rlvect_2(E,A)
=> ( ( v2_convex1(E,A)
& k10_rlvect_2(A,E) = k9_rlvect_2(A,B,C,D) )
=> ( B = C
| C = D
| D = B
| ( k3_real_1(k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,E,B),k8_funct_2(u1_struct_0(A),k1_numbers,E,C)),k8_funct_2(u1_struct_0(A),k1_numbers,E,D)) = np__1
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,B))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))
& r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))
& k13_rlvect_2(A,E) = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,k8_funct_2(u1_struct_0(A),k1_numbers,E,B)),k3_rlvect_1(A,C,k8_funct_2(u1_struct_0(A),k1_numbers,E,C))),k3_rlvect_1(A,D,k8_funct_2(u1_struct_0(A),k1_numbers,E,D))) ) ) ) ) ) ) ) ) ).
fof(d4_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k2_convex1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,C)
<=> ( v1_convex1(D,A)
& r1_tarski(B,D) ) ) ) ) ) ) ) ).
fof(d5_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_convex1(A,B) = k6_setfam_1(u1_struct_0(A),k2_convex1(A,B)) ) ) ).
fof(t30_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_convex1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,C)
=> r1_tarski(k3_convex1(A,B),C) ) ) ) ) ).
fof(t31_convex1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C,D] :
~ ( v2_funct_1(A)
& k2_relat_1(A) = k1_enumset1(B,C,D)
& B != C
& C != D
& D != B
& A != k11_finseq_1(B,C,D)
& A != k11_finseq_1(B,D,C)
& A != k11_finseq_1(C,B,D)
& A != k11_finseq_1(C,D,B)
& A != k11_finseq_1(D,B,C)
& A != k11_finseq_1(D,C,B) ) ) ).
fof(t32_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_convex1(A,B,np__1) = B ) ) ).
fof(t33_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_convex1(A,B,C) = k1_xboole_0 ) ) ) ).
fof(t34_convex1,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)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k1_convex1(A,B,np__0) = k7_rlvect_2(A,k1_rlvect_1(A)) ) ) ).
fof(t35_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_rusub_4(A,B,k7_rlvect_2(A,k1_rlvect_1(A))) = B ) ) ).
fof(t36_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k6_rusub_4(A,k6_rusub_4(A,B,C),D) = k6_rusub_4(A,B,k6_rusub_4(A,C,D)) ) ) ) ) ).
fof(t37_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k1_convex1(A,k1_convex1(A,B,D),C) = k1_convex1(A,B,k4_real_1(C,D)) ) ) ) ) ).
fof(t38_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k1_convex1(A,k6_rusub_4(A,B,C),D) = k6_rusub_4(A,k1_convex1(A,B,D),k1_convex1(A,C,D)) ) ) ) ) ).
fof(t39_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r1_tarski(B,C)
=> r1_tarski(k1_convex1(A,B,D),k1_convex1(A,C,D)) ) ) ) ) ) ).
fof(t40_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k6_rusub_4(A,B,C) = k1_xboole_0 ) ) ) ).
fof(dt_k1_convex1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_numbers) )
=> m1_subset_1(k1_convex1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_convex1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k2_convex1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k3_convex1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_convex1(k3_convex1(A,B),A)
& m1_subset_1(k3_convex1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(d1_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_convex1(A,B,C) = a_3_0_convex1(A,B,C) ) ) ) ).
fof(t17_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = a_3_1_convex1(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t18_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = a_3_2_convex1(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t19_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = a_3_3_convex1(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t20_convex1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = a_3_4_convex1(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(fraenkel_a_3_0_convex1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_0_convex1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k3_rlvect_1(B,E,D)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_1_convex1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_1_convex1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_xreal_0(D,k1_bhsp_1(B,E,C)) ) ) ) ).
fof(fraenkel_a_3_2_convex1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_2_convex1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ~ r1_xreal_0(k1_bhsp_1(B,E,C),D) ) ) ) ).
fof(fraenkel_a_3_3_convex1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_3_convex1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_xreal_0(k1_bhsp_1(B,E,C),D) ) ) ) ).
fof(fraenkel_a_3_4_convex1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_4_convex1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ~ r1_xreal_0(D,k1_bhsp_1(B,E,C)) ) ) ) ).
%------------------------------------------------------------------------------