SET007 Axioms: SET007+403.ax
%------------------------------------------------------------------------------
% File : SET007+403 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Extremal Properties of Vertices on Special Polygons, Part I
% 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 : sppol_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 74 ( 10 unt; 0 def)
% Number of atoms : 448 ( 70 equ)
% Maximal formula atoms : 19 ( 6 avg)
% Number of connectives : 445 ( 71 ~; 6 |; 161 &)
% ( 11 <=>; 196 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 9 con; 0-4 aty)
% Number of variables : 182 ( 178 !; 4 ?)
% 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(cc1_sppol_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( ~ v1_realset1(A)
& v1_sppol_1(A) )
=> ~ v2_sppol_1(A) ) ) ).
fof(cc2_sppol_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( ~ v1_realset1(A)
& v2_sppol_1(A) )
=> ~ v1_sppol_1(A) ) ) ).
fof(t1_sppol_1,axiom,
$true ).
fof(t2_sppol_1,axiom,
$true ).
fof(t3_sppol_1,axiom,
$true ).
fof(t4_sppol_1,axiom,
$true ).
fof(t5_sppol_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(k6_xcmplx_0(A,np__1),B)
& ~ r1_xreal_0(B,k6_xcmplx_0(A,np__1))
& r1_xreal_0(A,k2_xcmplx_0(B,np__1))
& ~ r1_xreal_0(k2_xcmplx_0(B,np__1),A) ) ) ) ) ).
fof(t6_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> r1_xreal_0(A,k5_real_1(B,np__1)) ) ) ) ).
fof(t7_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,k5_real_1(A,B))
& r1_xreal_0(k5_real_1(A,B),C) )
=> ( r2_hidden(k5_real_1(A,B),k2_finseq_1(C))
& m2_subset_1(k5_real_1(A,B),k1_numbers,k5_numbers) ) ) ) ) ) ).
fof(t8_sppol_1,axiom,
$true ).
fof(t9_sppol_1,axiom,
$true ).
fof(t10_sppol_1,axiom,
$true ).
fof(t11_sppol_1,axiom,
$true ).
fof(t12_sppol_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,np__1)
& r1_xreal_0(np__0,B)
& r1_xreal_0(np__0,C)
& k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(k6_xcmplx_0(np__1,A),C)) = np__0
& ~ ( A = np__0
& C = np__0 )
& ~ ( A = np__1
& B = np__0 )
& ~ ( B = np__0
& C = np__0 ) ) ) ) ) ).
fof(t13_sppol_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,A)
& r1_xreal_0(k1_square_1(B,C),A) ) ) ) ) ).
fof(t14_sppol_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(A,C)
& r1_xreal_0(A,k2_square_1(B,C)) ) ) ) ) ).
fof(t15_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ! [C] :
( m1_subset_1(C,k1_euclid(A))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> r1_xreal_0(k5_real_1(k12_euclid(k8_euclid(A,B,C)),k12_euclid(k8_euclid(A,C,D))),k12_euclid(k8_euclid(A,B,D))) ) ) ) ) ).
fof(t16_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ! [C] :
( m1_subset_1(C,k1_euclid(A))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> r1_xreal_0(k5_real_1(k12_euclid(k8_euclid(A,C,B)),k12_euclid(k8_euclid(A,C,D))),k12_euclid(k8_euclid(A,D,B))) ) ) ) ) ).
fof(t17_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ( m1_subset_1(B,k1_euclid(A))
& m1_subset_1(B,u1_struct_0(k14_euclid(A))) ) ) ) ).
fof(t18_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
=> ( m1_subset_1(B,k1_euclid(A))
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) ) ) ).
fof(t19_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) ) ) ).
fof(t20_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(A)))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ( ( D = B
& E = C )
=> k4_metric_1(k14_euclid(A),B,C) = k12_euclid(k8_euclid(A,D,E)) ) ) ) ) ) ) ).
fof(t21_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ~ ( r2_hidden(B,k3_topreal1(A,C,D))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1)
& B = k17_euclid(A,k18_euclid(k5_real_1(np__1,E),A,C),k18_euclid(E,A,D)) ) ) ) ) ) ) ) ).
fof(t22_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(np__0,D)
& r1_xreal_0(D,np__1) )
=> r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,D),A,B),k18_euclid(D,A,C)),k3_topreal1(A,B,C)) ) ) ) ) ) ).
fof(t23_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ~ ( v4_pre_topc(D,k15_euclid(A))
& r1_tarski(D,k3_topreal1(A,B,C))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,E),A,B),k18_euclid(E,A,C)),D)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(np__0,F)
& r1_xreal_0(F,np__1)
& r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,F),A,B),k18_euclid(F,A,C)),D) )
=> r1_xreal_0(E,F) ) ) ) ) ) ) ) ) ) ).
fof(t24_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C))
& r2_hidden(B,k3_topreal1(A,D,E))
& B != D
& B != E ) ) ) ) ) ) ).
fof(t25_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( k3_topreal1(A,B,C) = k3_topreal1(A,D,E)
& ~ ( B = D
& C = E )
& ~ ( B = E
& C = D ) ) ) ) ) ) ) ).
fof(t26_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v3_compts_1(k15_euclid(A)) ) ).
fof(t27_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> v4_pre_topc(k1_struct_0(k15_euclid(A),B),k15_euclid(A)) ) ) ).
fof(t28_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> v6_compts_1(k3_topreal1(A,B,C),k15_euclid(A)) ) ) ) ).
fof(t29_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> v4_pre_topc(k3_topreal1(A,B,C),k15_euclid(A)) ) ) ) ).
fof(d1_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_sppol_1(A,B,C)
<=> ( r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( r2_hidden(B,k3_topreal1(A,D,E))
& r1_tarski(k3_topreal1(A,D,E),C)
& B != D
& B != E ) ) ) ) ) ) ) ) ).
fof(t30_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( r1_sppol_1(A,B,C)
& r1_tarski(D,C)
& r2_hidden(B,D) )
=> r1_sppol_1(A,B,D) ) ) ) ) ) ).
fof(t31_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> r1_sppol_1(A,B,k1_struct_0(k15_euclid(A),B)) ) ) ).
fof(t32_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_sppol_1(A,B,k3_topreal1(A,B,C)) ) ) ) ).
fof(t33_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_sppol_1(A,B,k3_topreal1(A,C,B)) ) ) ) ).
fof(t34_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ~ ( r1_sppol_1(A,B,k3_topreal1(A,C,D))
& B != C
& B != D ) ) ) ) ) ).
fof(t35_sppol_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k21_euclid(A) != k21_euclid(B)
& k22_euclid(A) != k22_euclid(B)
& ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
& k21_euclid(C) != k21_euclid(A)
& k21_euclid(C) != k21_euclid(B)
& k22_euclid(C) != k22_euclid(A)
& k22_euclid(C) != k22_euclid(B) ) ) ) ) ) ).
fof(d2_sppol_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_sppol_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> k22_euclid(B) = k22_euclid(C) ) ) ) ) ) ).
fof(d3_sppol_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v2_sppol_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> k21_euclid(B) = k21_euclid(C) ) ) ) ) ) ).
fof(t36_sppol_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = k22_euclid(B)
<=> v1_sppol_1(k3_topreal1(np__2,A,B)) ) ) ) ).
fof(t37_sppol_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(A) = k21_euclid(B)
<=> v2_sppol_1(k3_topreal1(np__2,A,B)) ) ) ) ).
fof(t38_sppol_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(A,k3_topreal1(np__2,B,C))
& r2_hidden(D,k3_topreal1(np__2,B,C))
& k22_euclid(A) = k22_euclid(D) )
=> ( k21_euclid(A) = k21_euclid(D)
| v1_sppol_1(k3_topreal1(np__2,B,C)) ) ) ) ) ) ) ).
fof(t39_sppol_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(A,k3_topreal1(np__2,B,C))
& r2_hidden(D,k3_topreal1(np__2,B,C))
& k21_euclid(A) = k21_euclid(D) )
=> ( k22_euclid(A) = k22_euclid(D)
| v2_sppol_1(k3_topreal1(np__2,B,C)) ) ) ) ) ) ) ).
fof(t40_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> v4_pre_topc(k4_topreal1(np__2,B,A),k15_euclid(np__2)) ) ) ).
fof(t41_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& ~ v2_sppol_1(k4_topreal1(np__2,B,A))
& ~ v1_sppol_1(k4_topreal1(np__2,B,A)) ) ) ) ).
fof(t42_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v2_funct_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(B))
& v1_realset1(k4_topreal1(np__2,B,A)) ) ) ) ).
fof(t43_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v2_funct_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(B))
& v2_sppol_1(k4_topreal1(np__2,B,A))
& v1_sppol_1(k4_topreal1(np__2,B,A)) ) ) ) ).
fof(t49_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> v4_pre_topc(k5_topreal1(np__2,A),k15_euclid(np__2)) ) ).
fof(d4_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v3_sppol_1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__2),k3_finseq_1(A)) )
=> ( k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__2)))
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__2))) ) ) ) ) ) ).
fof(t50_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) )
=> k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))) ) ) ) ).
fof(t51_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) )
=> k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))) ) ) ) ).
fof(t52_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)
& D = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
& E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))
& ~ ( k21_euclid(C) = k21_euclid(D)
& k21_euclid(E) != k21_euclid(D) )
& ~ ( k22_euclid(C) = k22_euclid(D)
& k22_euclid(E) != k22_euclid(D) ) ) ) ) ) ) ) ).
fof(t53_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)
& D = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
& E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))
& ~ ( k21_euclid(D) = k21_euclid(E)
& k21_euclid(C) != k21_euclid(D) )
& ~ ( k22_euclid(D) = k22_euclid(E)
& k22_euclid(C) != k22_euclid(D) ) ) ) ) ) ) ) ).
fof(t54_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& r1_tarski(k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1)))) ) ) ) ).
fof(t55_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& v2_sppol_1(k4_topreal1(np__2,B,A)) )
=> v1_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) ).
fof(t56_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& v1_sppol_1(k4_topreal1(np__2,B,A)) )
=> v2_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) ).
fof(t57_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& ~ ( v2_sppol_1(k4_topreal1(np__2,B,A))
& v1_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) )
& ~ ( v1_sppol_1(k4_topreal1(np__2,B,A))
& v2_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) ) ).
fof(t58_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k3_topreal1(np__2,C,D))
& r1_tarski(k3_topreal1(np__2,C,D),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) != C
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) != D ) ) ) ) ) ).
fof(t59_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B)) )
=> r1_sppol_1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1)))) ) ) ) ).
fof(t60_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( v1_topreal1(B)
& v3_sppol_1(B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
& E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k3_topreal1(np__2,C,D)) )
=> ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) = D
| r2_hidden(C,k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
| ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ~ r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
& r2_hidden(G,k3_topreal1(np__2,C,D))
& r2_hidden(G,k9_metric_1(k14_euclid(np__2),E,F)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( r2_sppol_1(A,B,C)
<=> ( v3_sppol_1(A)
& v4_topreal1(A)
& v3_sppol_1(B)
& v4_topreal1(B)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))
& v3_sppol_1(k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__2)))
& v3_sppol_1(k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k5_real_1(k3_finseq_1(A),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k5_real_1(k3_finseq_1(B),np__1))))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)) = k2_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)))
& C = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) ) ) ).
fof(t61_sppol_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( r2_sppol_1(C,D,B)
=> ( r1_xreal_0(A,np__1)
| r1_xreal_0(k3_finseq_1(C),A)
| r1_sppol_1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A),B) ) ) ) ) ) ) ).
fof(t44_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> v1_finset_1(a_1_0_sppol_1(A)) ) ).
fof(t45_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> v1_finset_1(a_1_1_sppol_1(A)) ) ).
fof(t46_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(a_1_0_sppol_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) ).
fof(t47_sppol_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(a_1_1_sppol_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) ).
fof(t48_sppol_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( A = k3_tarski(a_1_1_sppol_1(B))
=> v4_pre_topc(A,k15_euclid(np__2)) ) ) ) ).
fof(s1_sppol_1,axiom,
v1_finset_1(a_0_0_sppol_1) ).
fof(s2_sppol_1,axiom,
v1_finset_1(a_0_1_sppol_1) ).
fof(fraenkel_a_1_0_sppol_1,axiom,
! [A,B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,a_1_0_sppol_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_topreal1(np__2,B,C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) ) ) ) ).
fof(fraenkel_a_1_1_sppol_1,axiom,
! [A,B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,a_1_1_sppol_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_topreal1(np__2,B,C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) ) ) ) ).
fof(fraenkel_a_0_0_sppol_1,axiom,
! [A] :
( r2_hidden(A,a_0_0_sppol_1)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s1_sppol_1(f2_s1_sppol_1,B)
& r2_hidden(B,k4_finseq_1(f2_s1_sppol_1))
& p1_s1_sppol_1(B) ) ) ).
fof(fraenkel_a_0_1_sppol_1,axiom,
! [A] :
( r2_hidden(A,a_0_1_sppol_1)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s2_sppol_1(f2_s2_sppol_1,B)
& r1_xreal_0(np__1,B)
& r1_xreal_0(B,k3_finseq_1(f2_s2_sppol_1))
& p1_s2_sppol_1(B) ) ) ).
%------------------------------------------------------------------------------