SET007 Axioms: SET007+445.ax
%------------------------------------------------------------------------------
% File : SET007+445 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Concept of the Triangulation
% 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 : triang_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 6 unt; 0 def)
% Number of atoms : 394 ( 38 equ)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 387 ( 57 ~; 3 |; 197 &)
% ( 11 <=>; 119 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-4 aty)
% Number of functors : 48 ( 48 usr; 9 con; 0-4 aty)
% Number of variables : 148 ( 134 !; 14 ?)
% 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_triang_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k1_finseq_1(k1_nat_1(A,np__1)))
& v1_finset_1(k1_finseq_1(k1_nat_1(A,np__1))) ) ) ).
fof(fc2_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ( ~ v3_struct_0(g1_orders_2(A,B))
& v1_orders_2(g1_orders_2(A,B))
& v2_orders_2(g1_orders_2(A,B))
& v3_orders_2(g1_orders_2(A,B))
& v4_orders_2(g1_orders_2(A,B)) ) ) ).
fof(rc1_triang_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
& ~ v1_xboole_0(B) ) ).
fof(rc2_triang_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
& ~ v1_xboole_0(B)
& v1_setfam_1(B) ) ) ).
fof(rc3_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_setfam_1(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ? [C] :
( m1_subset_1(C,B)
& ~ v1_xboole_0(C) ) ) ).
fof(rc4_triang_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
& ~ v1_xboole_0(B)
& ~ v2_setfam_1(B) ) ) ).
fof(rc5_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v2_setfam_1(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ? [C] :
( m1_subset_1(C,B)
& ~ v1_xboole_0(C)
& v1_finset_1(C) ) ) ).
fof(fc3_triang_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A)
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v1_finset_1(k3_orders_2(A,B,C)) ) ).
fof(fc4_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ( ~ v1_xboole_0(k3_triang_1(A))
& ~ v2_setfam_1(k3_triang_1(A)) ) ) ).
fof(cc1_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ! [C] :
( m1_subset_1(C,B)
=> v1_finset_1(C) ) ) ).
fof(rc6_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k3_triang_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc7_triang_1,axiom,
? [A] :
( m1_pboole(A,k5_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_triang_1(A) ) ).
fof(fc5_triang_1,axiom,
! [A,B] :
( ( v1_triang_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ~ v1_xboole_0(k1_funct_1(k4_triang_1(A),B)) ) ).
fof(fc6_triang_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k1_funct_1(k6_triang_1,A)) ) ).
fof(cc2_triang_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_funct_1(k6_triang_1,A))
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(rc8_triang_1,axiom,
? [A] :
( l1_triang_1(A)
& v2_triang_1(A) ) ).
fof(rc9_triang_1,axiom,
? [A] :
( l1_triang_1(A)
& v2_triang_1(A)
& v3_triang_1(A) ) ).
fof(fc7_triang_1,axiom,
! [A] :
( ( v3_triang_1(A)
& l1_triang_1(A) )
=> ( v1_relat_1(u1_triang_1(A))
& v1_funct_1(u1_triang_1(A))
& v1_triang_1(u1_triang_1(A)) ) ) ).
fof(fc8_triang_1,axiom,
! [A,B] :
( ( v1_triang_1(A)
& m1_pboole(A,k5_numbers)
& m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
=> ( v2_triang_1(g1_triang_1(A,B))
& v3_triang_1(g1_triang_1(A,B)) ) ) ).
fof(t1_triang_1,axiom,
! [A] : k1_toler_1(k1_xboole_0,A) = k1_xboole_0 ).
fof(t2_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_finset_1(B)
& B != k1_xboole_0
& ! [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)
& ~ r3_orders_2(A,C,D)
& ~ r3_orders_2(A,D,C) ) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r3_orders_2(A,C,D) ) ) ) ) ) ) ) ).
fof(t3_triang_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( ( r1_tarski(A,B)
& k4_card_1(A) = k4_card_1(B) )
=> A = B ) ) ) ).
fof(d1_triang_1,axiom,
$true ).
fof(d2_triang_1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r3_orders_1(C,B)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( D = k2_triang_1(A,B,C)
<=> ( k2_relat_1(D) = B
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k4_finseq_1(D))
& r2_hidden(F,k4_finseq_1(D)) )
=> ( r1_xreal_0(F,E)
| ( k4_finseq_4(k5_numbers,A,D,E) != k4_finseq_4(k5_numbers,A,D,F)
& r2_hidden(k4_tarski(k4_finseq_4(k5_numbers,A,D,E),k4_finseq_4(k5_numbers,A,D,F)),C) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_triang_1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( m2_finseq_1(D,A)
=> ( ( k2_relat_1(D) = B
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k4_finseq_1(D))
& r2_hidden(F,k4_finseq_1(D)) )
=> ( r1_xreal_0(F,E)
| ( k4_finseq_4(k5_numbers,A,D,E) != k4_finseq_4(k5_numbers,A,D,F)
& r2_hidden(k4_tarski(k4_finseq_4(k5_numbers,A,D,E),k4_finseq_4(k5_numbers,A,D,F)),C) ) ) ) ) ) )
=> D = k2_triang_1(A,B,C) ) ) ) ) ).
fof(t5_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(k1_struct_0(A,B),k3_triang_1(A)) ) ) ).
fof(t6_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> r2_hidden(k1_xboole_0,k3_triang_1(A)) ) ).
fof(t7_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B,C] :
( ( r1_tarski(B,C)
& r2_hidden(C,k3_triang_1(A)) )
=> r2_hidden(B,k3_triang_1(A)) ) ) ).
fof(t8_triang_1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r3_orders_1(C,B)
=> v2_funct_1(k2_triang_1(A,B,C)) ) ) ) ).
fof(t9_triang_1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r3_orders_1(C,B)
=> k3_finseq_1(k2_triang_1(A,B,C)) = k1_card_1(B) ) ) ) ).
fof(t10_triang_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_triang_1(C,u1_struct_0(B),k3_triang_1(B)) )
=> ( k1_card_1(C) = A
=> k4_finseq_1(k2_triang_1(u1_struct_0(B),C,u1_orders_2(B))) = k2_finseq_1(A) ) ) ) ) ).
fof(d4_triang_1,axiom,
! [A] :
( m1_pboole(A,k5_numbers)
=> ( v1_triang_1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ v1_xboole_0(k1_funct_1(A,B))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& v1_xboole_0(k1_funct_1(A,C)) ) ) ) ) ) ) ).
fof(d5_triang_1,axiom,
! [A] :
( m1_pboole(A,k5_numbers)
=> ! [B] :
( m1_pboole(B,k5_numbers)
=> ( B = k4_triang_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_funct_1(B,C) = k1_funct_2(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C)) ) ) ) ) ).
fof(d6_triang_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_fraenkel(B,k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1)),k1_fraenkel(k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1))))
=> k5_triang_1(A,B) = B ) ) ).
fof(d8_triang_1,axiom,
$true ).
fof(d9_triang_1,axiom,
! [A] :
( l1_triang_1(A)
=> ( v3_triang_1(A)
<=> v1_triang_1(u1_triang_1(A)) ) ) ).
fof(d10_triang_1,axiom,
! [A] :
( ( v3_triang_1(A)
& l1_triang_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k6_triang_1,B))
=> ( k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)) != k1_xboole_0
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u1_triang_1(A),B))
=> ( E = k7_triang_1(A,B,C,D)
<=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( ( F = k1_funct_1(u2_triang_1(A),B)
& G = k1_funct_1(F,D) )
=> E = k1_funct_1(G,C) ) ) ) ) ) ) ) ) ) ) ).
fof(s1_triang_1,axiom,
( ( p1_s1_triang_1(f1_s1_triang_1)
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( p1_s1_triang_1(k1_nat_1(A,np__1))
=> ( r1_xreal_0(f1_s1_triang_1,A)
| p1_s1_triang_1(A) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,f1_s1_triang_1)
=> p1_s1_triang_1(A) ) ) ) ).
fof(s2_triang_1,axiom,
( ( v1_finset_1(f2_s2_triang_1)
& p1_s2_triang_1(k1_subset_1(f1_s2_triang_1))
& ! [A] :
( m1_subset_1(A,f1_s2_triang_1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(f1_s2_triang_1))
=> ( ( r2_hidden(A,f2_s2_triang_1)
& r1_tarski(B,f2_s2_triang_1)
& p1_s2_triang_1(B) )
=> p1_s2_triang_1(k2_xboole_0(B,k1_tarski(A))) ) ) ) )
=> p1_s2_triang_1(f2_s2_triang_1) ) ).
fof(dt_m1_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ! [C] :
( m1_triang_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ? [C] : m1_triang_1(C,A,B) ) ).
fof(redefinition_m1_triang_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
=> ! [C] :
( m1_triang_1(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_l1_triang_1,axiom,
$true ).
fof(existence_l1_triang_1,axiom,
? [A] : l1_triang_1(A) ).
fof(abstractness_v2_triang_1,axiom,
! [A] :
( l1_triang_1(A)
=> ( v2_triang_1(A)
=> A = g1_triang_1(u1_triang_1(A),u2_triang_1(A)) ) ) ).
fof(dt_k1_triang_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( v1_relat_2(k1_triang_1(A,B,C))
& v4_relat_2(k1_triang_1(A,B,C))
& v8_relat_2(k1_triang_1(A,B,C))
& v1_partfun1(k1_triang_1(A,B,C),C,C)
& m2_relset_1(k1_triang_1(A,B,C),C,C) ) ) ).
fof(redefinition_k1_triang_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k1_triang_1(A,B,C) = k2_wellord1(B,C) ) ).
fof(dt_k2_triang_1,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> m2_finseq_1(k2_triang_1(A,B,C),A) ) ).
fof(dt_k3_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> m1_subset_1(k3_triang_1(A),k1_zfmisc_1(k5_finsub_1(u1_struct_0(A)))) ) ).
fof(dt_k4_triang_1,axiom,
! [A] :
( m1_pboole(A,k5_numbers)
=> m1_pboole(k4_triang_1(A),k5_numbers) ) ).
fof(dt_k5_triang_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_fraenkel(k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1)))) )
=> m2_finseq_1(k5_triang_1(A,B),k1_numbers) ) ).
fof(dt_k6_triang_1,axiom,
m1_pboole(k6_triang_1,k5_numbers) ).
fof(dt_k7_triang_1,axiom,
! [A,B,C,D] :
( ( v3_triang_1(A)
& l1_triang_1(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)))
& m1_subset_1(D,k1_funct_1(k6_triang_1,B)) )
=> m1_subset_1(k7_triang_1(A,B,C,D),k1_funct_1(u1_triang_1(A),B)) ) ).
fof(dt_k8_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ( v2_triang_1(k8_triang_1(A))
& v3_triang_1(k8_triang_1(A))
& l1_triang_1(k8_triang_1(A)) ) ) ).
fof(dt_u1_triang_1,axiom,
! [A] :
( l1_triang_1(A)
=> m1_pboole(u1_triang_1(A),k5_numbers) ) ).
fof(dt_u2_triang_1,axiom,
! [A] :
( l1_triang_1(A)
=> m3_pboole(u2_triang_1(A),k5_numbers,k6_triang_1,k4_triang_1(u1_triang_1(A))) ) ).
fof(dt_g1_triang_1,axiom,
! [A,B] :
( ( m1_pboole(A,k5_numbers)
& m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
=> ( v2_triang_1(g1_triang_1(A,B))
& l1_triang_1(g1_triang_1(A,B)) ) ) ).
fof(free_g1_triang_1,axiom,
! [A,B] :
( ( m1_pboole(A,k5_numbers)
& m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
=> ! [C,D] :
( g1_triang_1(A,B) = g1_triang_1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(d3_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> k3_triang_1(A) = a_1_0_triang_1(A) ) ).
fof(d7_triang_1,axiom,
! [A] :
( m1_pboole(A,k5_numbers)
=> ( A = k6_triang_1
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_funct_1(A,B) = a_1_1_triang_1(B) ) ) ) ).
fof(d11_triang_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v2_triang_1(B)
& v3_triang_1(B)
& l1_triang_1(B) )
=> ( B = k8_triang_1(A)
<=> ( k1_funct_1(u1_triang_1(B),np__0) = k1_tarski(k1_xboole_0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k1_funct_1(u1_triang_1(B),C) = a_2_0_triang_1(A,C) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k6_triang_1,C))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u1_triang_1(B),k1_nat_1(C,np__1)))
=> ( r2_hidden(E,k1_funct_1(u1_triang_1(B),k1_nat_1(C,np__1)))
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_triang_1(F,u1_struct_0(A),k3_triang_1(A)) )
=> ( k2_triang_1(u1_struct_0(A),F,u1_orders_2(A)) = E
=> k7_triang_1(B,C,E,D) = k5_relat_1(D,k2_triang_1(u1_struct_0(A),F,u1_orders_2(A))) ) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_1_0_triang_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B) )
=> ( r2_hidden(A,a_1_0_triang_1(B))
<=> ? [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(B)))
& A = C
& r3_orders_1(u1_orders_2(B),C) ) ) ) ).
fof(fraenkel_a_1_1_triang_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_1_triang_1(B))
<=> ? [C] :
( m2_fraenkel(C,k2_finseq_1(B),k2_finseq_1(k1_nat_1(B,np__1)),k1_fraenkel(k2_finseq_1(B),k2_finseq_1(k1_nat_1(B,np__1))))
& A = C
& v1_goboard1(k5_triang_1(B,C)) ) ) ) ).
fof(fraenkel_a_2_0_triang_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_triang_1(B,C))
<=> ? [D] :
( ~ v1_xboole_0(D)
& m1_triang_1(D,u1_struct_0(B),k3_triang_1(B))
& A = k2_triang_1(u1_struct_0(B),D,u1_orders_2(B))
& k1_card_1(D) = C ) ) ) ).
%------------------------------------------------------------------------------