SET007 Axioms: SET007+785.ax
%------------------------------------------------------------------------------
% File : SET007+785 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lines in n-Dimensional Euclidean Spaces
% 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 : euclid_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 89 ( 0 unt; 0 def)
% Number of atoms : 448 ( 81 equ)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 391 ( 32 ~; 2 |; 100 &)
% ( 10 <=>; 247 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 45 ( 45 usr; 6 con; 0-4 aty)
% Number of variables : 299 ( 291 !; 8 ?)
% 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_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ~ v1_xboole_0(k1_euclid_4(A,B,C)) ) ).
fof(fc2_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> ~ v1_xboole_0(k6_euclid_4(A,B,C)) ) ).
fof(t1_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ( k7_euclid(A,k9_euclid(A,np__0,B),B) = B
& k7_euclid(A,B,k5_euclid(A)) = B ) ) ) ).
fof(t2_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_euclid(B,A,k5_euclid(B)) = k5_euclid(B) ) ) ).
fof(t3_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ( k9_euclid(A,np__1,B) = B
& k9_euclid(A,np__0,B) = k5_euclid(A) ) ) ) ).
fof(t4_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_euclid(C))
=> k9_euclid(C,k4_real_1(A,B),D) = k9_euclid(C,A,k9_euclid(C,B,D)) ) ) ) ) ).
fof(t5_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_euclid(B))
=> ~ ( k9_euclid(B,A,C) = k5_euclid(B)
& A != np__0
& C != k5_euclid(B) ) ) ) ) ).
fof(t6_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_euclid(B))
=> ! [D] :
( m1_subset_1(D,k1_euclid(B))
=> k9_euclid(B,A,k7_euclid(B,C,D)) = k7_euclid(B,k9_euclid(B,A,C),k9_euclid(B,A,D)) ) ) ) ) ).
fof(t7_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_euclid(C))
=> k9_euclid(C,k3_real_1(A,B),D) = k7_euclid(C,k9_euclid(C,A,D),k9_euclid(C,B,D)) ) ) ) ) ).
fof(t8_euclid_4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_euclid(B))
=> ! [D] :
( m1_subset_1(D,k1_euclid(B))
=> ~ ( k9_euclid(B,A,C) = k9_euclid(B,A,D)
& A != np__0
& C != D ) ) ) ) ) ).
fof(t9_euclid_4,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))
=> k1_euclid_4(A,B,C) = k1_euclid_4(A,C,B) ) ) ) ).
fof(t10_euclid_4,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))
=> ( r2_hidden(B,k2_euclid_4(A,B,C))
& r2_hidden(C,k2_euclid_4(A,B,C)) ) ) ) ) ).
fof(t11_euclid_4,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))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ( ( r2_hidden(B,k2_euclid_4(A,C,D))
& r2_hidden(E,k2_euclid_4(A,C,D)) )
=> r1_tarski(k2_euclid_4(A,B,E),k2_euclid_4(A,C,D)) ) ) ) ) ) ) ).
fof(t12_euclid_4,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))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ( ( r2_hidden(B,k2_euclid_4(A,C,D))
& r2_hidden(E,k2_euclid_4(A,C,D)) )
=> ( B = E
| r1_tarski(k2_euclid_4(A,C,D),k2_euclid_4(A,B,E)) ) ) ) ) ) ) ) ).
fof(d2_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ( v1_euclid_4(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_euclid(A))
& ? [D] :
( m1_subset_1(D,k1_euclid(A))
& C != D
& B = k2_euclid_4(A,C,D) ) ) ) ) ) ).
fof(t13_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ~ ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(D,C)
& r2_hidden(E,C)
& D != E
& B != C ) ) ) ) ) ) ).
fof(t14_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ~ ( v1_euclid_4(B,A)
& ! [C] :
( m1_subset_1(C,k1_euclid(A))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D ) ) ) ) ) ) ).
fof(t15_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
=> ~ ( v1_euclid_4(C,A)
& ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ~ ( B != D
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(d3_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k3_euclid_4(A,B) = B ) ) ).
fof(d4_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k4_euclid_4(A,B) = k12_euclid(k3_euclid_4(A,B)) ) ) ).
fof(d5_euclid_4,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))
=> k5_euclid_4(A,B,C) = k1_euclid_2(k3_euclid_4(A,B),k3_euclid_4(A,C)) ) ) ) ).
fof(t16_euclid_4,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))
=> k5_euclid_4(A,B,C) = k4_real_1(k6_real_1(np__1,np__4),k5_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C))))) ) ) ) ).
fof(t17_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> r1_xreal_0(np__0,k5_euclid_4(A,B,B)) ) ) ).
fof(t18_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k7_square_1(k4_euclid_4(A,B)) = k5_euclid_4(A,B,B) ) ) ).
fof(t19_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> r1_xreal_0(np__0,k4_euclid_4(A,B)) ) ) ).
fof(t20_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k4_euclid_4(A,B) = k9_square_1(k5_euclid_4(A,B,B)) ) ) ).
fof(t21_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ( k5_euclid_4(A,B,B) = np__0
<=> k4_euclid_4(A,B) = np__0 ) ) ) ).
fof(t22_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> ( k5_euclid_4(A,B,B) = np__0
<=> B = k5_euclid(A) ) ) ) ).
fof(t23_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k5_euclid_4(A,B,k5_euclid(A)) = np__0 ) ) ).
fof(t24_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_euclid(A))
=> k5_euclid_4(A,k5_euclid(A),B) = np__0 ) ) ).
fof(t25_euclid_4,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))
=> k5_euclid_4(A,k7_euclid(A,B,C),D) = k3_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,C,D)) ) ) ) ) ).
fof(t26_euclid_4,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] :
( v1_xreal_0(D)
=> k5_euclid_4(A,k9_euclid(A,D,B),C) = k3_xcmplx_0(D,k5_euclid_4(A,B,C)) ) ) ) ) ).
fof(t27_euclid_4,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] :
( v1_xreal_0(D)
=> k5_euclid_4(A,B,k9_euclid(A,D,C)) = k3_xcmplx_0(D,k5_euclid_4(A,B,C)) ) ) ) ) ).
fof(t28_euclid_4,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))
=> k5_euclid_4(A,k6_euclid(A,B),C) = k1_real_1(k5_euclid_4(A,B,C)) ) ) ) ).
fof(t29_euclid_4,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))
=> k5_euclid_4(A,B,k6_euclid(A,C)) = k1_real_1(k5_euclid_4(A,B,C)) ) ) ) ).
fof(t30_euclid_4,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))
=> k5_euclid_4(A,k6_euclid(A,B),k6_euclid(A,C)) = k5_euclid_4(A,B,C) ) ) ) ).
fof(t31_euclid_4,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))
=> k5_euclid_4(A,k8_euclid(A,B,C),D) = k5_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,C,D)) ) ) ) ) ).
fof(t32_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ! [F] :
( m1_subset_1(F,k1_euclid(A))
=> k5_euclid_4(A,k7_euclid(A,k9_euclid(A,B,D),k9_euclid(A,C,E)),F) = k2_xcmplx_0(k3_xcmplx_0(B,k5_euclid_4(A,D,F)),k3_xcmplx_0(C,k5_euclid_4(A,E,F))) ) ) ) ) ) ) ).
fof(t33_euclid_4,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))
=> k5_euclid_4(A,B,k7_euclid(A,C,D)) = k3_real_1(k5_euclid_4(A,B,C),k5_euclid_4(A,B,D)) ) ) ) ) ).
fof(t34_euclid_4,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))
=> k5_euclid_4(A,B,k8_euclid(A,C,D)) = k5_real_1(k5_euclid_4(A,B,C),k5_euclid_4(A,B,D)) ) ) ) ) ).
fof(t35_euclid_4,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))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> k5_euclid_4(A,k7_euclid(A,B,C),k7_euclid(A,D,E)) = k3_real_1(k3_real_1(k3_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,B,E)),k5_euclid_4(A,C,D)),k5_euclid_4(A,C,E)) ) ) ) ) ) ).
fof(t36_euclid_4,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))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> k5_euclid_4(A,k8_euclid(A,B,C),k8_euclid(A,D,E)) = k3_real_1(k5_real_1(k5_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,B,E)),k5_euclid_4(A,C,D)),k5_euclid_4(A,C,E)) ) ) ) ) ) ).
fof(t37_euclid_4,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))
=> k5_euclid_4(A,k7_euclid(A,B,C),k7_euclid(A,B,C)) = k3_real_1(k3_real_1(k5_euclid_4(A,B,B),k4_real_1(np__2,k5_euclid_4(A,B,C))),k5_euclid_4(A,C,C)) ) ) ) ).
fof(t38_euclid_4,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))
=> k5_euclid_4(A,k8_euclid(A,B,C),k8_euclid(A,B,C)) = k3_real_1(k5_real_1(k5_euclid_4(A,B,B),k4_real_1(np__2,k5_euclid_4(A,B,C))),k5_euclid_4(A,C,C)) ) ) ) ).
fof(t39_euclid_4,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))
=> k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))) = k3_real_1(k3_real_1(k7_square_1(k4_euclid_4(A,B)),k4_real_1(np__2,k5_euclid_4(A,B,C))),k7_square_1(k4_euclid_4(A,C))) ) ) ) ).
fof(t40_euclid_4,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))
=> k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C))) = k3_real_1(k5_real_1(k7_square_1(k4_euclid_4(A,B)),k4_real_1(np__2,k5_euclid_4(A,B,C))),k7_square_1(k4_euclid_4(A,C))) ) ) ) ).
fof(t41_euclid_4,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))
=> k3_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C)))) = k4_real_1(np__2,k3_real_1(k7_square_1(k4_euclid_4(A,B)),k7_square_1(k4_euclid_4(A,C)))) ) ) ) ).
fof(t42_euclid_4,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))
=> k5_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C)))) = k4_real_1(np__4,k5_euclid_4(A,B,C)) ) ) ) ).
fof(t43_euclid_4,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))
=> r1_xreal_0(k18_complex1(k5_euclid_4(A,B,C)),k4_real_1(k4_euclid_4(A,B),k4_euclid_4(A,C))) ) ) ) ).
fof(t44_euclid_4,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))
=> r1_xreal_0(k4_euclid_4(A,k7_euclid(A,B,C)),k3_real_1(k4_euclid_4(A,B),k4_euclid_4(A,C))) ) ) ) ).
fof(d6_euclid_4,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))
=> ( r1_euclid_4(A,B,C)
<=> k5_euclid_4(A,B,C) = np__0 ) ) ) ) ).
fof(t46_euclid_4,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)))
=> k6_euclid_4(A,B,C) = k6_euclid_4(A,C,B) ) ) ) ).
fof(t47_euclid_4,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)))
=> ( r2_hidden(B,k7_euclid_4(A,B,C))
& r2_hidden(C,k7_euclid_4(A,B,C)) ) ) ) ) ).
fof(t48_euclid_4,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)))
=> ( ( r2_hidden(B,k7_euclid_4(A,C,D))
& r2_hidden(E,k7_euclid_4(A,C,D)) )
=> r1_tarski(k7_euclid_4(A,B,E),k7_euclid_4(A,C,D)) ) ) ) ) ) ) ).
fof(t49_euclid_4,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)))
=> ( ( r2_hidden(B,k7_euclid_4(A,C,D))
& r2_hidden(E,k7_euclid_4(A,C,D)) )
=> ( B = E
| r1_tarski(k7_euclid_4(A,C,D),k7_euclid_4(A,B,E)) ) ) ) ) ) ) ) ).
fof(d8_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( v2_euclid_4(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
& ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
& C != D
& B = k7_euclid_4(A,C,D) ) ) ) ) ) ).
fof(t50_euclid_4,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_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ~ ( v2_euclid_4(D,A)
& v2_euclid_4(E,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r2_hidden(B,E)
& r2_hidden(C,E)
& B != C
& D != E ) ) ) ) ) ) ).
fof(t51_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ~ ( v2_euclid_4(B,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(C,B)
& r2_hidden(D,B)
& C != D ) ) ) ) ) ) ).
fof(t52_euclid_4,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))))
=> ~ ( v2_euclid_4(C,A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ~ ( B != D
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(d9_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k8_euclid_4(A,B) = B ) ) ).
fof(d10_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k9_euclid_4(A,B) = k4_euclid_4(A,k8_euclid_4(A,B)) ) ) ).
fof(d11_euclid_4,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)))
=> k10_euclid_4(A,B,C) = k5_euclid_4(A,k8_euclid_4(A,B),k8_euclid_4(A,C)) ) ) ) ).
fof(d12_euclid_4,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)))
=> ( r2_euclid_4(A,B,C)
<=> k10_euclid_4(A,B,C) = np__0 ) ) ) ) ).
fof(symmetry_r1_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( r1_euclid_4(A,B,C)
=> r1_euclid_4(A,C,B) ) ) ).
fof(symmetry_r2_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> ( r2_euclid_4(A,B,C)
=> r2_euclid_4(A,C,B) ) ) ).
fof(dt_k1_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> m1_subset_1(k1_euclid_4(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) ).
fof(dt_k2_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> m1_subset_1(k2_euclid_4(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) ).
fof(commutativity_k2_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k2_euclid_4(A,B,C) = k2_euclid_4(A,C,B) ) ).
fof(redefinition_k2_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k2_euclid_4(A,B,C) = k1_euclid_4(A,B,C) ) ).
fof(dt_k3_euclid_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> m2_finseq_1(k3_euclid_4(A,B),k1_numbers) ) ).
fof(dt_k4_euclid_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> m1_subset_1(k4_euclid_4(A,B),k1_numbers) ) ).
fof(dt_k5_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> m1_subset_1(k5_euclid_4(A,B,C),k1_numbers) ) ).
fof(commutativity_k5_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k5_euclid_4(A,B,C) = k5_euclid_4(A,C,B) ) ).
fof(dt_k6_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k6_euclid_4(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k7_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k7_euclid_4(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(commutativity_k7_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k7_euclid_4(A,B,C) = k7_euclid_4(A,C,B) ) ).
fof(redefinition_k7_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k7_euclid_4(A,B,C) = k6_euclid_4(A,B,C) ) ).
fof(dt_k8_euclid_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k8_euclid_4(A,B),k1_euclid(A)) ) ).
fof(dt_k9_euclid_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k9_euclid_4(A,B),k1_numbers) ) ).
fof(dt_k10_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k10_euclid_4(A,B,C),k1_numbers) ) ).
fof(commutativity_k10_euclid_4,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k10_euclid_4(A,B,C) = k10_euclid_4(A,C,B) ) ).
fof(d1_euclid_4,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))
=> k1_euclid_4(A,B,C) = a_3_0_euclid_4(A,B,C) ) ) ) ).
fof(t45_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ! [C] :
( m1_subset_1(C,k1_euclid(A))
=> ! [D] :
( m1_subset_1(D,k1_euclid(A))
=> ! [E] :
( m1_subset_1(E,k1_euclid(A))
=> ~ ( B = a_4_0_euclid_4(A,C,D,E)
& ! [F] :
( m1_subset_1(F,k1_euclid(A))
=> ~ ( r2_hidden(F,k2_euclid_4(A,C,D))
& k4_euclid_4(A,k8_euclid(A,E,F)) = k5_seq_4(B)
& r1_euclid_4(A,k8_euclid(A,C,D),k8_euclid(A,E,F)) ) ) ) ) ) ) ) ) ).
fof(d7_euclid_4,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)))
=> k6_euclid_4(A,B,C) = a_3_1_euclid_4(A,B,C) ) ) ) ).
fof(t53_euclid_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ! [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)))
=> ~ ( B = a_4_1_euclid_4(A,C,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ~ ( r2_hidden(F,k7_euclid_4(A,C,D))
& k9_euclid_4(A,k20_euclid(A,E,F)) = k5_seq_4(B)
& r2_euclid_4(A,k20_euclid(A,C,D),k20_euclid(A,E,F)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_euclid_4,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,k1_euclid(B))
& m1_subset_1(D,k1_euclid(B)) )
=> ( r2_hidden(A,a_3_0_euclid_4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& A = k7_euclid(B,k9_euclid(B,k5_real_1(np__1,E),C),k9_euclid(B,E,D)) ) ) ) ).
fof(fraenkel_a_4_0_euclid_4,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,k1_euclid(B))
& m1_subset_1(D,k1_euclid(B))
& m1_subset_1(E,k1_euclid(B)) )
=> ( r2_hidden(A,a_4_0_euclid_4(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_euclid(B))
& A = k4_euclid_4(B,k8_euclid(B,E,F))
& r2_hidden(F,k2_euclid_4(B,C,D)) ) ) ) ).
fof(fraenkel_a_3_1_euclid_4,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& m1_subset_1(D,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_3_1_euclid_4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& A = k17_euclid(B,k18_euclid(k5_real_1(np__1,E),B,C),k18_euclid(E,B,D)) ) ) ) ).
fof(fraenkel_a_4_1_euclid_4,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& m1_subset_1(D,u1_struct_0(k15_euclid(B)))
& m1_subset_1(E,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_4_1_euclid_4(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = k9_euclid_4(B,k20_euclid(B,E,F))
& r2_hidden(F,k7_euclid_4(B,C,D)) ) ) ) ).
%------------------------------------------------------------------------------