SET007 Axioms: SET007+758.ax
%------------------------------------------------------------------------------
% File : SET007+758 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Chains on a Grating in Euclidean Space
% 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 : chain_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 137 ( 2 unt; 0 def)
% Number of atoms : 1152 ( 124 equ)
% Maximal formula atoms : 37 ( 8 avg)
% Number of connectives : 1272 ( 257 ~; 19 |; 480 &)
% ( 53 <=>; 463 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 44 ( 42 usr; 1 prp; 0-5 aty)
% Number of functors : 63 ( 63 usr; 17 con; 0-5 aty)
% Number of variables : 557 ( 513 !; 44 ?)
% 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_chain_1,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_finset_1(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A)
& v1_rat_1(A) ) ).
fof(fc1_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers) )
=> ( ~ v1_xboole_0(k1_finseq_1(A))
& v1_finset_1(k1_finseq_1(A)) ) ) ).
fof(rc2_chain_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& ~ v1_realset1(A) ) ).
fof(fc2_chain_1,axiom,
! [A,B] :
( ~ v1_realset1(A)
=> ( ~ v1_xboole_0(k2_xboole_0(A,B))
& ~ v1_realset1(k2_xboole_0(A,B)) ) ) ).
fof(fc3_chain_1,axiom,
! [A,B] :
( ~ v1_realset1(A)
=> ( ~ v1_xboole_0(k2_xboole_0(B,A))
& ~ v1_realset1(k2_xboole_0(B,A)) ) ) ).
fof(fc4_chain_1,axiom,
( ~ v1_xboole_0(k1_numbers)
& ~ v1_realset1(k1_numbers)
& v1_membered(k1_numbers)
& v2_membered(k1_numbers) ) ).
fof(rc3_chain_1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& ~ v1_realset1(B) ) ) ).
fof(t1_chain_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t2_chain_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(C,B) ) ) ) ).
fof(d1_chain_1,axiom,
! [A] :
( ( v1_xreal_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ( v1_xboole_0(A)
<=> r1_xreal_0(A,np__0) ) ) ).
fof(d2_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_xboole_0(A)
<=> ~ r1_xreal_0(np__1,A) ) ) ).
fof(d3_chain_1,axiom,
! [A] :
( v1_realset1(A)
<=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> B = C ) ) ).
fof(t3_chain_1,axiom,
$true ).
fof(t4_chain_1,axiom,
! [A,B] :
( v1_realset1(k2_tarski(A,B))
<=> A = B ) ).
fof(t5_chain_1,axiom,
! [A,B] :
~ ( v1_realset1(A)
& ~ v1_realset1(k2_xboole_0(A,k1_tarski(B)))
& ! [C] : A != k1_tarski(C) ) ).
fof(t6_chain_1,axiom,
! [A] :
( k1_card_1(A) = np__2
<=> ? [B,C] :
( r2_hidden(B,A)
& r2_hidden(C,A)
& B != C
& ! [D] :
~ ( r2_hidden(D,A)
& D != B
& D != C ) ) ) ).
fof(t7_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( v1_abian(A)
<=> v1_abian(B) )
<=> v1_abian(k1_nat_1(A,B)) ) ) ) ).
fof(t8_chain_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r1_xboole_0(A,B)
=> ( ( v1_abian(k4_card_1(A))
<=> v1_abian(k4_card_1(B)) )
<=> v1_abian(k4_card_1(k2_xboole_0(A,B))) ) ) ) ) ).
fof(t9_chain_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( ( v1_abian(k4_card_1(A))
<=> v1_abian(k4_card_1(B)) )
<=> v1_abian(k4_card_1(k5_xboole_0(A,B))) ) ) ) ).
fof(d4_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,k1_numbers) )
=> ( B = k1_euclid(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_funct_1(C)
& v1_funct_2(C,k2_finseq_1(A),k1_numbers)
& m2_relset_1(C,k2_finseq_1(A),k1_numbers) ) ) ) ) ) ).
fof(d5_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers))
& m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) )
=> ( m1_chain_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k2_finseq_1(A))
=> ( ~ v1_realset1(k8_funct_2(k2_finseq_1(A),k1_zfmisc_1(k1_numbers),B,C))
& v1_finset_1(k8_funct_2(k2_finseq_1(A),k1_zfmisc_1(k1_numbers),B,C)) ) ) ) ) ) ).
fof(t10_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m1_chain_1(C,A)
=> ( r2_hidden(B,k4_card_3(C))
<=> ! [D] :
( m1_subset_1(D,k2_finseq_1(A))
=> r2_hidden(k2_chain_1(A,B,D),k3_chain_1(A,C,D)) ) ) ) ) ) ).
fof(t11_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> v1_finset_1(k4_card_3(B)) ) ) ).
fof(t12_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) ) ) ) ).
fof(t13_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) ) ) ) ).
fof(t14_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(B,A)
& r2_hidden(C,A)
& ~ r1_xreal_0(C,B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(C,D) ) ) ) ) ) ).
fof(t15_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) ) ) ) ).
fof(t16_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(B,A)
& r2_hidden(C,A)
& ~ r1_xreal_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( r1_xreal_0(C,D)
& r1_xreal_0(D,B) ) ) ) ) ) ) ).
fof(d6_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers))
=> ( m2_chain_1(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& B = k1_domain_1(k1_numbers,k1_numbers,C,D)
& r2_hidden(C,A)
& r2_hidden(D,A)
& ( ( ~ r1_xreal_0(D,C)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,A)
& ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(D,E) ) ) )
| ( ~ r1_xreal_0(C,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> ( r1_xreal_0(E,C)
& r1_xreal_0(D,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
<=> ( r2_hidden(B,A)
& r2_hidden(C,A)
& ( ( ~ r1_xreal_0(C,B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(C,D) ) ) )
| ( ~ r1_xreal_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( r1_xreal_0(D,B)
& r1_xreal_0(C,D) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( A = k7_domain_1(k1_numbers,B,C)
=> ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,D,E),A)
<=> ( ( D = B
& E = C )
| ( D = C
& E = B ) ) ) ) ) ) ) ) ) ).
fof(t19_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ).
fof(t20_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ).
fof(t21_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,D),A) )
=> C = D ) ) ) ) ) ).
fof(t22_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,D,C),A) )
=> B = D ) ) ) ) ) ).
fof(t23_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,C,B),A)
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,E,D),A) )
=> ( r1_xreal_0(C,B)
| r1_xreal_0(E,D)
| ( C = E
& B = D ) ) ) ) ) ) ) ) ).
fof(t24_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( r2_hidden(B,k4_chain_1(A,C,D))
<=> ~ ( ~ ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ( r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
& r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E)) ) )
& ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ~ ( ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,D,E))
& ( r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E))
| r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) ) ) ) ) ) ) ) ) ) ).
fof(t25_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) )
=> ( r2_hidden(D,k4_chain_1(A,B,C))
<=> ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ( r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E))
& r1_xreal_0(k2_chain_1(A,D,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) ) ).
fof(t26_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( ~ ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) )
=> ( r2_hidden(D,k4_chain_1(A,C,B))
<=> ? [E] :
( m1_subset_1(E,k2_finseq_1(A))
& ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
& ( r1_xreal_0(k2_chain_1(A,D,E),k2_chain_1(A,B,E))
| r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,D,E)) ) ) ) ) ) ) ) ) ).
fof(t27_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r2_hidden(B,k4_chain_1(A,B,C))
& r2_hidden(C,k4_chain_1(A,B,C)) ) ) ) ) ).
fof(t28_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k4_chain_1(A,B,B) = k6_domain_1(k1_euclid(A),B) ) ) ).
fof(t29_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
=> ( r1_tarski(k4_chain_1(A,D,E),k4_chain_1(A,B,C))
<=> ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ( r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,D,F))
& r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
& r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,C,F)) ) ) ) ) ) ) ) ) ) ).
fof(t30_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,C,F),k2_chain_1(A,B,F)) )
=> ( r1_tarski(k4_chain_1(A,C,B),k4_chain_1(A,D,E))
<=> ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ( r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,E,F))
& ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
& r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,C,F)) ) ) ) ) ) ) ) ) ) ).
fof(t31_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( ( ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
& ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F)) ) )
=> ( r1_tarski(k4_chain_1(A,B,C),k4_chain_1(A,E,D))
<=> ~ ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ( ~ r1_xreal_0(k2_chain_1(A,C,F),k2_chain_1(A,D,F))
& ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,B,F)) ) ) ) ) ) ) ) ) ) ).
fof(t32_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( ( ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
| ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) ) )
=> ( k4_chain_1(A,B,C) = k4_chain_1(A,D,E)
<=> ( B = D
& C = E ) ) ) ) ) ) ) ) ).
fof(t33_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( r1_xreal_0(A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_euclid(B)))
=> ( r2_hidden(D,k5_chain_1(B,C,A))
<=> ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
& D = k4_chain_1(B,E,F)
& ~ ( ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(B)))
=> ~ ( k4_card_1(G) = A
& ! [H] :
( m1_subset_1(H,k2_finseq_1(B))
=> ( ( r2_hidden(H,G)
& ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) )
| ( ~ r2_hidden(H,G)
& k2_chain_1(B,E,H) = k2_chain_1(B,F,H)
& r2_hidden(k2_chain_1(B,E,H),k3_chain_1(B,C,H)) ) ) ) ) )
& ~ ( A = B
& ! [G] :
( m1_subset_1(G,k2_finseq_1(B))
=> ( ~ r1_xreal_0(k2_chain_1(B,E,G),k2_chain_1(B,F,G))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,G),k2_chain_1(B,F,G)),k3_chain_1(B,C,G)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t34_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m1_chain_1(E,B)
=> ( r1_xreal_0(A,B)
=> ( r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
<=> ~ ( ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k2_finseq_1(B)))
=> ~ ( k4_card_1(F) = A
& ! [G] :
( m1_subset_1(G,k2_finseq_1(B))
=> ( ( r2_hidden(G,F)
& ~ r1_xreal_0(k2_chain_1(B,D,G),k2_chain_1(B,C,G))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,G),k2_chain_1(B,D,G)),k3_chain_1(B,E,G)) )
| ( ~ r2_hidden(G,F)
& k2_chain_1(B,C,G) = k2_chain_1(B,D,G)
& r2_hidden(k2_chain_1(B,C,G),k3_chain_1(B,E,G)) ) ) ) ) )
& ~ ( A = B
& ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t35_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m1_chain_1(E,B)
=> ~ ( r1_xreal_0(A,B)
& r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
& ? [F] :
( m1_subset_1(F,k2_finseq_1(B))
& ~ ( ~ r1_xreal_0(k2_chain_1(B,D,F),k2_chain_1(B,C,F))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) )
& ~ ( k2_chain_1(B,C,F) = k2_chain_1(B,D,F)
& r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F)) ) )
& ~ ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) ).
fof(t36_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m1_chain_1(E,B)
=> ( ( r1_xreal_0(A,B)
& r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A)) )
=> ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> ( r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F))
& r2_hidden(k2_chain_1(B,D,F),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) ).
fof(t37_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m1_chain_1(E,B)
=> ~ ( r1_xreal_0(A,B)
& r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
& ~ ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) )
& ? [F] :
( m1_subset_1(F,k2_finseq_1(B))
& r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) ) ) ) ) ) ) ) ).
fof(t38_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
=> ( r2_hidden(C,k5_chain_1(A,B,np__0))
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& C = k4_chain_1(A,D,D)
& ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> r2_hidden(k2_chain_1(A,D,E),k3_chain_1(A,B,E)) ) ) ) ) ) ) ).
fof(t39_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_chain_1(D,A)
=> ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,np__0))
<=> ( B = C
& ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> r2_hidden(k2_chain_1(A,B,E),k3_chain_1(A,D,E)) ) ) ) ) ) ) ) ).
fof(t40_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
=> ( r2_hidden(C,k5_chain_1(A,B,A))
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& C = k4_chain_1(A,D,E)
& ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F)) )
& ( ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F)) )
| ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F)) ) ) ) ) ) ) ) ) ).
fof(t41_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_chain_1(D,A)
=> ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,A))
<=> ( ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E)) )
& ( ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) )
| ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) ) ).
fof(t42_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( B = k1_nat_1(A,np__1)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_euclid(B)))
=> ( r2_hidden(D,k5_chain_1(B,C,A))
<=> ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
& ? [G] :
( m1_subset_1(G,k2_finseq_1(B))
& D = k4_chain_1(B,E,F)
& k2_chain_1(B,E,G) = k2_chain_1(B,F,G)
& r2_hidden(k2_chain_1(B,E,G),k3_chain_1(B,C,G))
& ! [H] :
( m1_subset_1(H,k2_finseq_1(B))
=> ( H != G
=> ( ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t43_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m1_chain_1(E,B)
=> ( B = k1_nat_1(A,np__1)
=> ( r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
<=> ? [F] :
( m1_subset_1(F,k2_finseq_1(B))
& k2_chain_1(B,C,F) = k2_chain_1(B,D,F)
& r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F))
& ! [G] :
( m1_subset_1(G,k2_finseq_1(B))
=> ( G != F
=> ( ~ r1_xreal_0(k2_chain_1(B,D,G),k2_chain_1(B,C,G))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,G),k2_chain_1(B,D,G)),k3_chain_1(B,E,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t44_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
=> ( r2_hidden(C,k5_chain_1(A,B,np__1))
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& ? [F] :
( m1_subset_1(F,k2_finseq_1(A))
& C = k4_chain_1(A,D,E)
& ~ ( r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F))
& ~ ( A = np__1
& ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F)) ) )
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F))
& ! [G] :
( m1_subset_1(G,k2_finseq_1(A))
=> ( G != F
=> ( k2_chain_1(A,D,G) = k2_chain_1(A,E,G)
& r2_hidden(k2_chain_1(A,D,G),k3_chain_1(A,B,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t45_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_chain_1(D,A)
=> ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,np__1))
<=> ? [E] :
( m1_subset_1(E,k2_finseq_1(A))
& ~ ( r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
& ~ ( A = np__1
& ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) )
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E))
& ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ( F != E
=> ( k2_chain_1(A,B,F) = k2_chain_1(A,C,F)
& r2_hidden(k2_chain_1(A,B,F),k3_chain_1(A,D,F)) ) ) ) ) ) ) ) ) ) ).
fof(t46_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(C))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(C))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(C))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(C))
=> ! [H] :
( m1_chain_1(H,C)
=> ( ( r1_xreal_0(A,C)
& r1_xreal_0(B,C)
& r2_hidden(k4_chain_1(C,D,E),k5_chain_1(C,H,A))
& r2_hidden(k4_chain_1(C,F,G),k5_chain_1(C,H,B))
& r1_tarski(k4_chain_1(C,D,E),k4_chain_1(C,F,G)) )
=> ! [I] :
( m1_subset_1(I,k2_finseq_1(C))
=> ~ ( ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
& k2_chain_1(C,E,I) = k2_chain_1(C,G,I) )
& ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
& k2_chain_1(C,E,I) = k2_chain_1(C,F,I) )
& ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,G,I)
& k2_chain_1(C,E,I) = k2_chain_1(C,G,I) )
& ~ ( r1_xreal_0(k2_chain_1(C,D,I),k2_chain_1(C,E,I))
& ~ r1_xreal_0(k2_chain_1(C,F,I),k2_chain_1(C,G,I))
& r1_xreal_0(k2_chain_1(C,G,I),k2_chain_1(C,D,I))
& r1_xreal_0(k2_chain_1(C,E,I),k2_chain_1(C,F,I)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t47_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(C))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(C))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(C))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(C))
=> ! [H] :
( m1_chain_1(H,C)
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(B,C)
& r2_hidden(k4_chain_1(C,D,E),k5_chain_1(C,H,A))
& r2_hidden(k4_chain_1(C,F,G),k5_chain_1(C,H,B))
& r1_tarski(k4_chain_1(C,D,E),k4_chain_1(C,F,G))
& ! [I] :
( m1_subset_1(I,k2_finseq_1(C))
=> ( ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
& k2_chain_1(C,E,I) = k2_chain_1(C,F,I) )
& ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,G,I)
& k2_chain_1(C,E,I) = k2_chain_1(C,G,I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t48_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m1_chain_1(F,A)
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(k2_finseq_1(A)))
=> ( ( r1_tarski(k4_chain_1(A,B,C),k4_chain_1(A,D,E))
& ! [I] :
( m1_subset_1(I,k2_finseq_1(A))
=> ( ( r2_hidden(I,G)
& ~ r1_xreal_0(k2_chain_1(A,C,I),k2_chain_1(A,B,I))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,I),k2_chain_1(A,C,I)),k3_chain_1(A,F,I)) )
| ( ~ r2_hidden(I,G)
& k2_chain_1(A,B,I) = k2_chain_1(A,C,I)
& r2_hidden(k2_chain_1(A,B,I),k3_chain_1(A,F,I)) ) ) )
& ! [I] :
( m1_subset_1(I,k2_finseq_1(A))
=> ( ( r2_hidden(I,H)
& ~ r1_xreal_0(k2_chain_1(A,E,I),k2_chain_1(A,D,I))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,I),k2_chain_1(A,E,I)),k3_chain_1(A,F,I)) )
| ( ~ r2_hidden(I,H)
& k2_chain_1(A,D,I) = k2_chain_1(A,E,I)
& r2_hidden(k2_chain_1(A,D,I),k3_chain_1(A,F,I)) ) ) ) )
=> ( r1_tarski(G,H)
& ! [I] :
( m1_subset_1(I,k2_finseq_1(A))
=> ( ~ ( ~ r2_hidden(I,G)
& r2_hidden(I,H) )
=> ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I)
& k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) )
& ! [I] :
( m1_subset_1(I,k2_finseq_1(A))
=> ~ ( ~ r2_hidden(I,G)
& r2_hidden(I,H)
& ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I)
& k2_chain_1(A,C,I) = k2_chain_1(A,D,I) )
& ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,E,I)
& k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k6_chain_1(A,B,C) = k1_xboole_0 ) ) ) ).
fof(d10_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> k7_chain_1(A,B) = k5_chain_1(A,B,A) ) ) ).
fof(d11_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,A))
=> ( C = k9_chain_1(A,B)
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& C = k4_chain_1(A,D,E)
& ! [F] :
( m1_subset_1(F,k2_finseq_1(A))
=> ( ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F)) ) ) ) ) ) ) ) ) ).
fof(t49_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_chain_1(D,A)
=> ( m2_subset_1(k4_chain_1(A,B,C),k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,D,A))
=> ( k4_chain_1(A,B,C) = k9_chain_1(A,D)
<=> ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) ).
fof(t50_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m1_chain_1(D,A)
=> ( k4_chain_1(A,B,C) = k9_chain_1(A,D)
<=> ! [E] :
( m1_subset_1(E,k2_finseq_1(A))
=> ( ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E)) ) ) ) ) ) ) ) ).
fof(t51_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,np__1)))
=> ( r2_hidden(E,k10_chain_1(B,C,A,D))
<=> r1_tarski(D,E) ) ) ) ) ) ) ).
fof(d14_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C)))
=> ( r1_chain_1(A,B,C,D,E)
<=> E = k11_chain_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t52_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
=> ( r2_hidden(D,k11_chain_1(B,C,A,E))
<=> ( r1_xreal_0(k1_nat_1(A,np__1),B)
& ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(A,np__1)),k10_chain_1(B,C,A,D),E))) ) ) ) ) ) ) ) ).
fof(t53_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( ~ r1_xreal_0(k1_nat_1(A,np__1),B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
=> k11_chain_1(B,C,A,D) = k6_chain_1(B,C,A) ) ) ) ) ) ).
fof(t54_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( r1_xreal_0(k1_nat_1(A,np__1),B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,np__1)))
=> ( r2_hidden(D,k11_chain_1(B,C,A,k6_domain_1(k5_chain_1(B,C,k1_nat_1(A,np__1)),E)))
<=> r1_tarski(D,E) ) ) ) ) ) ) ) ).
fof(t55_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( B = k1_nat_1(A,np__1)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
=> k4_card_1(k10_chain_1(B,C,A,D)) = np__2 ) ) ) ) ) ).
fof(t56_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,k1_nat_1(np__0,np__1)))
=> k4_card_1(k11_chain_1(A,B,np__0,k6_domain_1(k5_chain_1(A,B,k1_nat_1(np__0,np__1)),C))) = np__2 ) ) ) ).
fof(t57_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ( k7_chain_1(A,B) = k3_subset_1(k5_chain_1(A,B,A),k6_chain_1(A,B,A))
& k6_chain_1(A,B,A) = k3_subset_1(k5_chain_1(A,B,A),k7_chain_1(A,B)) ) ) ) ).
fof(t58_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
=> k8_chain_1(B,C,A,D,k6_chain_1(B,C,A)) = D ) ) ) ) ).
fof(t59_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
=> k8_chain_1(B,C,A,D,D) = k6_chain_1(B,C,A) ) ) ) ) ).
fof(t60_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(A,B,A)))
=> k3_subset_1(k5_chain_1(A,B,A),C) = k8_chain_1(A,B,A,C,k7_chain_1(A,B)) ) ) ) ).
fof(t61_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> k11_chain_1(B,C,A,k6_chain_1(B,C,k1_nat_1(A,np__1))) = k6_chain_1(B,C,A) ) ) ) ).
fof(t62_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_chain_1(B,k1_nat_1(A,np__1))
=> k11_chain_1(k1_nat_1(A,np__1),B,A,k7_chain_1(k1_nat_1(A,np__1),B)) = k6_chain_1(k1_nat_1(A,np__1),B,A) ) ) ).
fof(t63_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
=> k11_chain_1(B,C,A,k8_chain_1(B,C,k1_nat_1(A,np__1),D,E)) = k8_chain_1(B,C,A,k11_chain_1(B,C,A,D),k11_chain_1(B,C,A,E)) ) ) ) ) ) ).
fof(t64_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_chain_1(B,k1_nat_1(A,np__1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(k1_nat_1(A,np__1),B,k1_nat_1(A,np__1))))
=> k11_chain_1(k1_nat_1(A,np__1),B,A,k3_subset_1(k5_chain_1(k1_nat_1(A,np__1),B,k1_nat_1(A,np__1)),C)) = k11_chain_1(k1_nat_1(A,np__1),B,A,C) ) ) ) ).
fof(t65_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,np__1),np__1))))
=> k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,np__1),D)) = k6_chain_1(B,C,A) ) ) ) ) ).
fof(d15_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
=> ( m3_chain_1(D,A,B,C)
<=> ~ ( ~ ( C = np__0
& v1_abian(k4_card_1(D)) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( C = k1_nat_1(E,np__1)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(E,np__1))))
& F = D
& k11_chain_1(A,B,E,F) = k6_chain_1(A,B,E) ) ) ) ) ) ) ) ) ) ).
fof(t66_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
=> ( m3_chain_1(D,B,C,k1_nat_1(A,np__1))
<=> k11_chain_1(B,C,A,D) = k6_chain_1(B,C,A) ) ) ) ) ) ).
fof(t67_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ( ~ r1_xreal_0(A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
=> m3_chain_1(D,B,C,A) ) ) ) ) ) ).
fof(t68_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(A,B,np__0)))
=> ( m3_chain_1(C,A,B,np__0)
<=> v1_abian(k4_card_1(C)) ) ) ) ) ).
fof(d16_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m3_chain_1(D,A,B,k1_nat_1(C,np__1))
=> k11_chain_1(A,B,C,D) = k6_chain_1(A,B,C) ) ) ) ) ).
fof(t69_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m3_chain_1(C,A,B,A)
=> m3_chain_1(k3_subset_1(k5_chain_1(A,B,A),C),A,B,A) ) ) ) ).
fof(d17_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( ~ v3_struct_0(D)
& v1_rlvect_1(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& l1_rlvect_1(D) )
=> ( D = k16_chain_1(A,B,C)
<=> ( u1_struct_0(D) = k1_chain_1(k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,C))
& k1_rlvect_1(D) = k12_chain_1(A,B,C)
& ! [E] :
( m1_subset_1(E,u1_struct_0(D))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(D))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k5_chain_1(A,B,C)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(k5_chain_1(A,B,C)))
=> ( ( E = G
& F = H )
=> k4_rlvect_1(D,E,F) = k8_chain_1(A,B,C,G,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t70_chain_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_chain_1(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
<=> m1_subset_1(D,u1_struct_0(k16_chain_1(B,C,A))) ) ) ) ) ).
fof(d18_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_mod_4(D,k16_chain_1(A,B,k1_nat_1(C,np__1)),k16_chain_1(A,B,C))
=> ( D = k17_chain_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k16_chain_1(A,B,k1_nat_1(C,np__1))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
=> ( E = F
=> k8_funct_2(u1_struct_0(k16_chain_1(A,B,k1_nat_1(C,np__1))),u1_struct_0(k16_chain_1(A,B,C)),D,E) = k15_chain_1(A,B,C,F) ) ) ) ) ) ) ) ) ).
fof(s2_chain_1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s2_chain_1)
=> ( r2_hidden(A,f2_s2_chain_1)
=> p1_s2_chain_1(k6_domain_1(f1_s2_chain_1,A)) ) )
& ! [A] :
( m1_subset_1(A,f1_s2_chain_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(f1_s2_chain_1)) )
=> ( ( r2_hidden(A,f2_s2_chain_1)
& r1_tarski(B,f2_s2_chain_1)
& p1_s2_chain_1(B) )
=> ( r2_hidden(A,B)
| p1_s2_chain_1(k4_subset_1(f1_s2_chain_1,B,k6_domain_1(f1_s2_chain_1,A))) ) ) ) ) )
=> p1_s2_chain_1(f2_s2_chain_1) ) ).
fof(s3_chain_1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s3_chain_1)
=> ! [B] :
( m1_subset_1(B,f1_s3_chain_1)
=> ( ( r2_hidden(A,f2_s3_chain_1)
& r2_hidden(B,f2_s3_chain_1) )
=> ( A = B
| p1_s3_chain_1(k7_domain_1(f1_s3_chain_1,A,B)) ) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s3_chain_1)
=> ! [B] :
( ( v1_finset_1(B)
& ~ v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(f1_s3_chain_1)) )
=> ( ( r2_hidden(A,f2_s3_chain_1)
& r1_tarski(B,f2_s3_chain_1)
& p1_s3_chain_1(B) )
=> ( r2_hidden(A,B)
| p1_s3_chain_1(k4_subset_1(f1_s3_chain_1,B,k6_domain_1(f1_s3_chain_1,A))) ) ) ) ) )
=> p1_s3_chain_1(f2_s3_chain_1) ) ).
fof(s4_chain_1,axiom,
( ( p1_s4_chain_1(k6_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1))
& ! [A] :
( m2_subset_1(A,k1_zfmisc_1(k1_euclid(f1_s4_chain_1)),k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1))
=> ( r2_hidden(A,f4_s4_chain_1)
=> p1_s4_chain_1(k6_domain_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1),A)) ) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1)))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1)))
=> ( ( r1_tarski(A,f4_s4_chain_1)
& r1_tarski(B,f4_s4_chain_1)
& p1_s4_chain_1(A)
& p1_s4_chain_1(B) )
=> p1_s4_chain_1(k8_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1,A,B)) ) ) ) )
=> p1_s4_chain_1(f4_s4_chain_1) ) ).
fof(dt_m1_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers))
& m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) ) ) ) ).
fof(existence_m1_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers) )
=> ? [B] : m1_chain_1(B,A) ) ).
fof(dt_m2_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m2_chain_1(B,A)
=> m1_subset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers)) ) ) ).
fof(existence_m2_chain_1,axiom,
! [A] :
( ( v1_finset_1(A)
& ~ v1_realset1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] : m2_chain_1(B,A) ) ).
fof(dt_m3_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m3_chain_1(D,A,B,C)
=> m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) ) ) ).
fof(existence_m3_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ? [D] : m3_chain_1(D,A,B,C) ) ).
fof(dt_k1_chain_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> m1_subset_1(k1_chain_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(redefinition_k1_chain_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k1_chain_1(A,B) = k1_zfmisc_1(B) ) ).
fof(dt_k2_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k2_finseq_1(A)) )
=> m1_subset_1(k2_chain_1(A,B,C),k1_numbers) ) ).
fof(redefinition_k2_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k2_finseq_1(A)) )
=> k2_chain_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k3_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k2_finseq_1(A)) )
=> ( v1_finset_1(k3_chain_1(A,B,C))
& ~ v1_realset1(k3_chain_1(A,B,C))
& m1_subset_1(k3_chain_1(A,B,C),k1_zfmisc_1(k1_numbers)) ) ) ).
fof(redefinition_k3_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k2_finseq_1(A)) )
=> k3_chain_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k4_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( ~ v1_xboole_0(k4_chain_1(A,B,C))
& m1_subset_1(k4_chain_1(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) ) ).
fof(dt_k5_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( ~ v1_xboole_0(k5_chain_1(A,B,C))
& v1_finset_1(k5_chain_1(A,B,C))
& m1_subset_1(k5_chain_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(k1_euclid(A)))) ) ) ).
fof(dt_k6_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k6_chain_1(A,B,C),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ).
fof(dt_k7_chain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A) )
=> m1_subset_1(k7_chain_1(A,B),k1_zfmisc_1(k5_chain_1(A,B,A))) ) ).
fof(dt_k8_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
& m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
=> m1_subset_1(k8_chain_1(A,B,C,D,E),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ).
fof(commutativity_k8_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
& m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
=> k8_chain_1(A,B,C,D,E) = k8_chain_1(A,B,C,E,D) ) ).
fof(redefinition_k8_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
& m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
=> k8_chain_1(A,B,C,D,E) = k5_xboole_0(D,E) ) ).
fof(dt_k9_chain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A) )
=> m2_subset_1(k9_chain_1(A,B),k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,A)) ) ).
fof(dt_k10_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_chain_1(A,B,C)) )
=> m1_subset_1(k10_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) ) ).
fof(dt_k11_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
=> m1_subset_1(k11_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ).
fof(dt_k12_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m3_chain_1(k12_chain_1(A,B,C),A,B,C) ) ).
fof(redefinition_k12_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> k12_chain_1(A,B,C) = k6_chain_1(A,B,C) ) ).
fof(dt_k13_chain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A) )
=> m3_chain_1(k13_chain_1(A,B),A,B,A) ) ).
fof(redefinition_k13_chain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A) )
=> k13_chain_1(A,B) = k7_chain_1(A,B) ) ).
fof(dt_k14_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m3_chain_1(D,A,B,C)
& m3_chain_1(E,A,B,C) )
=> m3_chain_1(k14_chain_1(A,B,C,D,E),A,B,C) ) ).
fof(commutativity_k14_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m3_chain_1(D,A,B,C)
& m3_chain_1(E,A,B,C) )
=> k14_chain_1(A,B,C,D,E) = k14_chain_1(A,B,C,E,D) ) ).
fof(redefinition_k14_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m3_chain_1(D,A,B,C)
& m3_chain_1(E,A,B,C) )
=> k14_chain_1(A,B,C,D,E) = k5_xboole_0(D,E) ) ).
fof(dt_k15_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
=> m3_chain_1(k15_chain_1(A,B,C,D),A,B,C) ) ).
fof(redefinition_k15_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
=> k15_chain_1(A,B,C,D) = k11_chain_1(A,B,C,D) ) ).
fof(dt_k16_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( ~ v3_struct_0(k16_chain_1(A,B,C))
& v1_rlvect_1(k16_chain_1(A,B,C))
& v3_rlvect_1(k16_chain_1(A,B,C))
& v4_rlvect_1(k16_chain_1(A,B,C))
& v5_rlvect_1(k16_chain_1(A,B,C))
& v6_rlvect_1(k16_chain_1(A,B,C))
& l1_rlvect_1(k16_chain_1(A,B,C)) ) ) ).
fof(dt_k17_chain_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_chain_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m1_mod_4(k17_chain_1(A,B,C),k16_chain_1(A,B,k1_nat_1(C,np__1)),k16_chain_1(A,B,C)) ) ).
fof(d7_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> k4_chain_1(A,B,C) = a_3_0_chain_1(A,B,C) ) ) ) ).
fof(d8_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,A)
=> k5_chain_1(A,B,C) = a_3_1_chain_1(A,B,C) ) ) ) ) ).
fof(d12_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,C))
=> k10_chain_1(A,B,C,D) = a_4_0_chain_1(A,B,C,D) ) ) ) ) ).
fof(d13_chain_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_chain_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
=> k11_chain_1(A,B,C,D) = a_4_1_chain_1(A,B,C,D) ) ) ) ) ).
fof(s1_chain_1,axiom,
r1_tarski(a_0_0_chain_1,f1_s1_chain_1) ).
fof(fraenkel_a_3_0_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& m2_finseq_2(C,k1_numbers,k1_euclid(B))
& m2_finseq_2(D,k1_numbers,k1_euclid(B)) )
=> ( r2_hidden(A,a_3_0_chain_1(B,C,D))
<=> ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& A = E
& ~ ( ~ ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> ( r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F))
& r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F)) ) )
& ! [F] :
( m1_subset_1(F,k2_finseq_1(B))
=> ~ ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
& ( r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F))
| r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_1_chain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& m1_chain_1(C,B)
& m2_subset_1(D,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_3_1_chain_1(B,C,D))
<=> ? [E,F] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& m2_finseq_2(F,k1_numbers,k1_euclid(B))
& A = k4_chain_1(B,E,F)
& ~ ( ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(B)))
=> ~ ( k4_card_1(G) = D
& ! [H] :
( m1_subset_1(H,k2_finseq_1(B))
=> ( ( r2_hidden(H,G)
& ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) )
| ( ~ r2_hidden(H,G)
& k2_chain_1(B,E,H) = k2_chain_1(B,F,H)
& r2_hidden(k2_chain_1(B,E,H),k3_chain_1(B,C,H)) ) ) ) ) )
& ~ ( D = B
& ! [G] :
( m1_subset_1(G,k2_finseq_1(B))
=> ( ~ r1_xreal_0(k2_chain_1(B,E,G),k2_chain_1(B,F,G))
& m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,G),k2_chain_1(B,F,G)),k3_chain_1(B,C,G)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_4_0_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& m1_chain_1(C,B)
& m2_subset_1(D,k1_numbers,k5_numbers)
& m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D)) )
=> ( r2_hidden(A,a_4_0_chain_1(B,C,D,E))
<=> ? [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(D,np__1)))
& A = F
& r1_tarski(E,F) ) ) ) ).
fof(fraenkel_a_4_1_chain_1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& m1_chain_1(C,B)
& m2_subset_1(D,k1_numbers,k5_numbers)
& m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(D,np__1)))) )
=> ( r2_hidden(A,a_4_1_chain_1(B,C,D,E))
<=> ? [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D))
& A = F
& r1_xreal_0(k1_nat_1(D,np__1),B)
& ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(D,np__1)),k10_chain_1(B,C,D,F),E))) ) ) ) ).
fof(fraenkel_a_0_0_chain_1,axiom,
! [A] :
( r2_hidden(A,a_0_0_chain_1)
<=> ? [B,C] :
( m1_subset_1(B,f2_s1_chain_1)
& m1_subset_1(C,f2_s1_chain_1)
& A = f3_s1_chain_1(B,C)
& p1_s1_chain_1(B,C) ) ) ).
%------------------------------------------------------------------------------