SET007 Axioms: SET007+775.ax
%------------------------------------------------------------------------------
% File : SET007+775 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Angle and Triangle in Euclidian Topological 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 : euclid_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 102 ( 3 unt; 0 def)
% Number of atoms : 553 ( 155 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 521 ( 70 ~; 6 |; 163 &)
% ( 19 <=>; 263 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 8 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 73 ( 73 usr; 11 con; 0-5 aty)
% Number of variables : 304 ( 281 !; 23 ?)
% 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(d1_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k1_euclid_3(A) = k23_euclid(k3_complex1(A),k4_complex1(A)) ) ).
fof(d2_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k2_euclid_3(A) = k5_arytm_0(k21_euclid(A),k22_euclid(A)) ) ).
fof(t1_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k2_euclid_3(k1_euclid_3(A)) = A ) ).
fof(t2_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k1_euclid_3(k2_euclid_3(A)) = A ) ).
fof(t3_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ? [B] :
( m1_subset_1(B,k2_numbers)
& A = k1_euclid_3(B) ) ) ).
fof(t4_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = k2_euclid_3(B) ) ) ).
fof(t5_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k1_euclid_3(A) = k1_euclid_3(B)
=> A = B ) ) ) ).
fof(t6_euclid_3,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)))
=> ( k2_euclid_3(A) = k2_euclid_3(B)
=> A = B ) ) ) ).
fof(t7_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k21_euclid(k1_euclid_3(A)) = k3_complex1(A)
& k22_euclid(k1_euclid_3(A)) = k4_complex1(A) ) ) ).
fof(t8_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k3_complex1(k2_euclid_3(A)) = k21_euclid(A)
& k4_complex1(k2_euclid_3(A)) = k22_euclid(A) ) ) ).
fof(t9_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k1_euclid_3(k5_arytm_0(A,B)) = k23_euclid(A,B) ) ) ).
fof(t10_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k23_euclid(k3_complex1(k8_complex1(A,B)),k4_complex1(k8_complex1(A,B))) = k23_euclid(k3_real_1(k3_complex1(A),k3_complex1(B)),k3_real_1(k4_complex1(A),k4_complex1(B))) ) ) ).
fof(t11_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k1_euclid_3(k8_complex1(A,B)) = k17_euclid(np__2,k1_euclid_3(A),k1_euclid_3(B)) ) ) ).
fof(t12_euclid_3,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)))
=> k5_arytm_0(k21_euclid(k17_euclid(np__2,A,B)),k22_euclid(k17_euclid(np__2,A,B))) = k5_arytm_0(k3_real_1(k21_euclid(A),k21_euclid(B)),k3_real_1(k22_euclid(A),k22_euclid(B))) ) ) ).
fof(t13_euclid_3,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)))
=> k2_euclid_3(k17_euclid(np__2,A,B)) = k8_complex1(k2_euclid_3(A),k2_euclid_3(B)) ) ) ).
fof(t14_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k23_euclid(k3_complex1(k10_complex1(A)),k4_complex1(k10_complex1(A))) = k23_euclid(k1_real_1(k3_complex1(A)),k1_real_1(k4_complex1(A))) ) ).
fof(t15_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k1_euclid_3(k10_complex1(A)) = k19_euclid(np__2,k1_euclid_3(A)) ) ).
fof(t16_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k5_arytm_0(k21_euclid(k19_euclid(np__2,A)),k22_euclid(k19_euclid(np__2,A))) = k5_arytm_0(k1_real_1(k21_euclid(A)),k1_real_1(k22_euclid(A))) ) ).
fof(t17_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k2_euclid_3(k19_euclid(np__2,A)) = k10_complex1(k2_euclid_3(A)) ) ).
fof(t18_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k1_euclid_3(k11_complex1(A,B)) = k20_euclid(np__2,k1_euclid_3(A),k1_euclid_3(B)) ) ) ).
fof(t19_euclid_3,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)))
=> k2_euclid_3(k20_euclid(np__2,A,B)) = k11_complex1(k2_euclid_3(A),k2_euclid_3(B)) ) ) ).
fof(t20_euclid_3,axiom,
k1_euclid_3(k5_complex1) = k16_euclid(np__2) ).
fof(t21_euclid_3,axiom,
k2_euclid_3(k16_euclid(np__2)) = k5_complex1 ).
fof(t22_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k2_euclid_3(A) = k5_complex1
=> A = k16_euclid(np__2) ) ) ).
fof(t23_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k1_euclid_3(k9_complex1(k5_arytm_0(B,np__0),A)) = k18_euclid(B,np__2,k1_euclid_3(A)) ) ) ).
fof(t24_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k3_xcmplx_0(k2_xcmplx_0(A,k3_xcmplx_0(np__0,k7_complex1)),k2_xcmplx_0(B,k3_xcmplx_0(C,k7_complex1))) = k2_xcmplx_0(k4_real_1(A,B),k3_xcmplx_0(k4_real_1(A,C),k7_complex1)) ) ) ) ).
fof(t25_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k2_euclid_3(k18_euclid(A,np__2,B)) = k9_complex1(k5_arytm_0(A,np__0),k2_euclid_3(B)) ) ) ).
fof(t26_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k17_complex1(k2_euclid_3(A)) = k9_square_1(k3_real_1(k7_square_1(k21_euclid(A)),k7_square_1(k22_euclid(A)))) ) ).
fof(t27_euclid_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k3_finseq_1(A) = np__2
=> k12_euclid(A) = k9_square_1(k3_real_1(k7_square_1(k2_seq_1(k5_numbers,k1_numbers,A,np__1)),k7_square_1(k2_seq_1(k5_numbers,k1_numbers,A,np__2)))) ) ) ).
fof(t28_euclid_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( k3_finseq_1(A) = np__2
& B = A )
=> k5_toprns_1(np__2,B) = k12_euclid(A) ) ) ) ).
fof(t29_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k5_toprns_1(np__2,k1_euclid_3(A)) = k9_square_1(k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))) ) ).
fof(t30_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k5_toprns_1(np__2,k1_euclid_3(A)) = k17_complex1(A) ) ).
fof(t31_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k17_complex1(k2_euclid_3(A)) = k5_toprns_1(np__2,A) ) ).
fof(d3_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k3_euclid_3(A) = k1_comptrig(k2_euclid_3(A)) ) ).
fof(t32_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( A = k2_euclid_3(B)
| B = k1_euclid_3(A) )
=> k1_comptrig(A) = k3_euclid_3(B) ) ) ) ).
fof(t33_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k3_euclid_3(A))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k3_euclid_3(A)) ) ) ).
fof(t34_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( A = k4_real_1(k5_toprns_1(np__2,C),k23_sin_cos(k3_euclid_3(C)))
& B = k4_real_1(k5_toprns_1(np__2,C),k20_sin_cos(k3_euclid_3(C))) )
=> C = k23_euclid(A,B) ) ) ) ) ).
fof(t35_euclid_3,axiom,
k3_euclid_3(k16_euclid(np__2)) = np__0 ).
fof(t36_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( A != k16_euclid(np__2)
=> ( ( ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(A))
=> k3_euclid_3(k19_euclid(np__2,A)) = k3_real_1(k3_euclid_3(A),k32_sin_cos) )
& ( r1_xreal_0(k32_sin_cos,k3_euclid_3(A))
=> k3_euclid_3(k19_euclid(np__2,A)) = k5_real_1(k3_euclid_3(A),k32_sin_cos) ) ) ) ) ).
fof(t37_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k3_euclid_3(A) = np__0
=> ( A = k23_euclid(k5_toprns_1(np__2,A),np__0)
& k22_euclid(A) = np__0 ) ) ) ).
fof(t38_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( A != k16_euclid(np__2)
=> ( ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(A))
<=> r1_xreal_0(k32_sin_cos,k3_euclid_3(k19_euclid(np__2,A))) ) ) ) ).
fof(t39_euclid_3,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)))
=> ( ~ ( A = B
& k20_euclid(np__2,A,B) = k16_euclid(np__2) )
=> ( ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(k20_euclid(np__2,A,B)))
<=> r1_xreal_0(k32_sin_cos,k3_euclid_3(k20_euclid(np__2,B,A))) ) ) ) ) ).
fof(t40_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(k3_euclid_3(A),k2_rcomp_1(np__0,k32_sin_cos))
<=> ~ r1_xreal_0(k22_euclid(A),np__0) ) ) ).
fof(t41_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k3_euclid_3(A) != np__0
=> ( ~ ( ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(A))
& r1_xreal_0(k20_sin_cos(k3_euclid_3(A)),np__0) )
& ~ ( ~ r1_xreal_0(k20_sin_cos(k3_euclid_3(A)),np__0)
& r1_xreal_0(k32_sin_cos,k3_euclid_3(A)) ) ) ) ) ).
fof(t42_euclid_3,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)))
=> ~ ( ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(A))
& ~ r1_xreal_0(k32_sin_cos,k3_euclid_3(B))
& r1_xreal_0(k32_sin_cos,k3_euclid_3(k17_euclid(np__2,A,B))) ) ) ) ).
fof(d4_euclid_3,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)))
=> k4_euclid_3(A,B,C) = k5_complex2(k2_euclid_3(A),k2_euclid_3(B),k2_euclid_3(C)) ) ) ) ).
fof(t43_euclid_3,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)))
=> ( r1_xreal_0(np__0,k4_euclid_3(A,B,C))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k4_euclid_3(A,B,C)) ) ) ) ) ).
fof(t44_euclid_3,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)))
=> k4_euclid_3(A,B,C) = k4_euclid_3(k20_euclid(np__2,A,B),k16_euclid(np__2),k20_euclid(np__2,C,B)) ) ) ) ).
fof(t45_euclid_3,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)))
=> ( k4_euclid_3(A,B,C) = np__0
=> ( k3_euclid_3(k20_euclid(np__2,A,B)) = k3_euclid_3(k20_euclid(np__2,C,B))
& k4_euclid_3(C,B,A) = np__0 ) ) ) ) ) ).
fof(t46_euclid_3,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)))
=> ( k4_euclid_3(A,B,C) != np__0
=> k4_euclid_3(C,B,A) = k5_real_1(k4_real_1(np__2,k32_sin_cos),k4_euclid_3(A,B,C)) ) ) ) ) ).
fof(t47_euclid_3,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)))
=> ( k4_euclid_3(C,B,A) != np__0
=> k4_euclid_3(C,B,A) = k5_real_1(k4_real_1(np__2,k32_sin_cos),k4_euclid_3(A,B,C)) ) ) ) ) ).
fof(t48_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k3_complex1(k2_complex2(A,B)) = k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B))) ) ) ).
fof(t49_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_complex1(k2_complex2(A,B)) = k3_real_1(k1_real_1(k4_real_1(k3_complex1(A),k4_complex1(B))),k4_real_1(k4_complex1(A),k3_complex1(B))) ) ) ).
fof(t50_euclid_3,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)))
=> k2_euclid_2(np__2,A,B) = k3_real_1(k4_real_1(k21_euclid(A),k21_euclid(B)),k4_real_1(k22_euclid(A),k22_euclid(B))) ) ) ).
fof(t51_euclid_3,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)))
=> k2_euclid_2(np__2,A,B) = k3_complex1(k2_complex2(k2_euclid_3(A),k2_euclid_3(B))) ) ) ).
fof(t52_euclid_3,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)))
=> ~ ( A != k16_euclid(np__2)
& B != k16_euclid(np__2)
& ~ ( k2_euclid_2(np__2,A,B) = np__0
<=> ( k4_euclid_3(A,k16_euclid(np__2),B) = k6_real_1(k32_sin_cos,np__2)
| k4_euclid_3(A,k16_euclid(np__2),B) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) ) ) ) ) ) ) ).
fof(t53_euclid_3,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)))
=> ~ ( A != k16_euclid(np__2)
& B != k16_euclid(np__2)
& ~ ( ~ ( ( k3_real_1(k1_real_1(k4_real_1(k21_euclid(A),k22_euclid(B))),k4_real_1(k22_euclid(A),k21_euclid(B))) = k4_real_1(k5_toprns_1(np__2,A),k5_toprns_1(np__2,B))
| k3_real_1(k1_real_1(k4_real_1(k21_euclid(A),k22_euclid(B))),k4_real_1(k22_euclid(A),k21_euclid(B))) = k1_real_1(k4_real_1(k5_toprns_1(np__2,A),k5_toprns_1(np__2,B))) )
& k4_euclid_3(A,k16_euclid(np__2),B) != k6_real_1(k32_sin_cos,np__2)
& k4_euclid_3(A,k16_euclid(np__2),B) != k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& ~ ( ( k4_euclid_3(A,k16_euclid(np__2),B) = k6_real_1(k32_sin_cos,np__2)
| k4_euclid_3(A,k16_euclid(np__2),B) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& k3_real_1(k1_real_1(k4_real_1(k21_euclid(A),k22_euclid(B))),k4_real_1(k22_euclid(A),k21_euclid(B))) != k4_real_1(k5_toprns_1(np__2,A),k5_toprns_1(np__2,B))
& k3_real_1(k1_real_1(k4_real_1(k21_euclid(A),k22_euclid(B))),k4_real_1(k22_euclid(A),k21_euclid(B))) != k1_real_1(k4_real_1(k5_toprns_1(np__2,A),k5_toprns_1(np__2,B))) ) ) ) ) ) ).
fof(t54_euclid_3,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)))
=> ~ ( A != B
& C != B
& ~ ( k2_euclid_2(np__2,k20_euclid(np__2,A,B),k20_euclid(np__2,C,B)) = np__0
<=> ( k4_euclid_3(A,B,C) = k6_real_1(k32_sin_cos,np__2)
| k4_euclid_3(A,B,C) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) ) ) ) ) ) ) ).
fof(t55_euclid_3,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)))
=> ~ ( A != B
& C != B
& ( k4_euclid_3(A,B,C) = k6_real_1(k32_sin_cos,np__2)
| k4_euclid_3(A,B,C) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& k3_real_1(k7_square_1(k5_toprns_1(np__2,k20_euclid(np__2,A,B))),k7_square_1(k5_toprns_1(np__2,k20_euclid(np__2,C,B)))) != k7_square_1(k5_toprns_1(np__2,k20_euclid(np__2,A,C))) ) ) ) ) ).
fof(t56_euclid_3,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)))
=> ~ ( B != A
& A != C
& C != B
& ~ r1_xreal_0(k32_sin_cos,k4_euclid_3(B,A,C))
& ~ r1_xreal_0(k32_sin_cos,k4_euclid_3(A,C,B))
& ~ r1_xreal_0(k32_sin_cos,k4_euclid_3(C,B,A))
& k3_real_1(k3_real_1(k4_euclid_3(B,A,C),k4_euclid_3(A,C,B)),k4_euclid_3(C,B,A)) != k32_sin_cos ) ) ) ) ).
fof(d5_euclid_3,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)))
=> k5_euclid_3(A,B,C,D) = k4_subset_1(u1_struct_0(k15_euclid(A)),k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D)),k3_topreal1(A,D,B)) ) ) ) ) ).
fof(d7_euclid_3,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)))
=> k7_euclid_3(A,B,C,D) = k6_subset_1(u1_struct_0(k15_euclid(A)),k6_euclid_3(A,B,C,D),k5_euclid_3(A,B,C,D)) ) ) ) ) ).
fof(d9_euclid_3,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)))
=> k9_euclid_3(A,B,C,D) = k4_subset_1(u1_struct_0(k15_euclid(A)),k8_euclid_3(A,B,C,D),k6_euclid_3(A,B,C,D)) ) ) ) ) ).
fof(t57_euclid_3,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(E,k9_euclid_3(A,B,C,D))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( k3_real_1(k3_real_1(F,G),H) = np__1
& E = k17_euclid(A,k17_euclid(A,k18_euclid(F,A,B),k18_euclid(G,A,C)),k18_euclid(H,A,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(t58_euclid_3,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_tarski(k5_euclid_3(A,B,C,D),k6_euclid_3(A,B,C,D)) ) ) ) ) ).
fof(d10_euclid_3,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_euclid_3(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( k17_euclid(A,k18_euclid(D,A,B),k18_euclid(E,A,C)) = k16_euclid(A)
=> ( D = np__0
& E = np__0 ) ) ) ) ) ) ) ) ).
fof(t59_euclid_3,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_euclid_3(A,B,C)
=> ( B != C
& B != k16_euclid(A)
& C != k16_euclid(A) ) ) ) ) ) ).
fof(t60_euclid_3,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_euclid_3(A,k20_euclid(A,C,B),k20_euclid(A,D,B))
& r2_hidden(E,k9_euclid_3(A,B,C,D))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( E = k17_euclid(A,k17_euclid(A,k18_euclid(F,A,B),k18_euclid(G,A,C)),k18_euclid(H,A,D))
& k3_real_1(k3_real_1(F,G),H) = np__1
& ! [I] :
( m1_subset_1(I,k1_numbers)
=> ! [J] :
( m1_subset_1(J,k1_numbers)
=> ! [K] :
( m1_subset_1(K,k1_numbers)
=> ( ( E = k17_euclid(A,k17_euclid(A,k18_euclid(I,A,B),k18_euclid(J,A,C)),k18_euclid(K,A,D))
& k3_real_1(k3_real_1(I,J),K) = np__1 )
=> ( I = F
& J = G
& K = H ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t61_euclid_3,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)))
=> ( ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& E = k17_euclid(A,k17_euclid(A,k18_euclid(F,A,B),k18_euclid(G,A,C)),k18_euclid(H,A,D))
& k3_real_1(k3_real_1(F,G),H) = np__1 ) ) )
=> r2_hidden(E,k9_euclid_3(A,B,C,D)) ) ) ) ) ) ) ).
fof(t63_euclid_3,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)))
=> ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
=> k9_euclid_3(np__2,A,B,C) = k1_euclid(np__2) ) ) ) ) ).
fof(d11_euclid_3,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_euclid_3(A,k20_euclid(A,C,B),k20_euclid(A,D,B))
& r2_hidden(E,k9_euclid_3(A,B,C,D)) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( F = k10_euclid_3(A,B,C,D,E)
<=> ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& k3_real_1(k3_real_1(F,G),H) = np__1
& E = k17_euclid(A,k17_euclid(A,k18_euclid(F,A,B),k18_euclid(G,A,C)),k18_euclid(H,A,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(d12_euclid_3,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_euclid_3(A,k20_euclid(A,C,B),k20_euclid(A,D,B))
& r2_hidden(E,k9_euclid_3(A,B,C,D)) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( F = k11_euclid_3(A,B,C,D,E)
<=> ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& k3_real_1(k3_real_1(G,F),H) = np__1
& E = k17_euclid(A,k17_euclid(A,k18_euclid(G,A,B),k18_euclid(F,A,C)),k18_euclid(H,A,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(d13_euclid_3,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_euclid_3(A,k20_euclid(A,C,B),k20_euclid(A,D,B))
& r2_hidden(E,k9_euclid_3(A,B,C,D)) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( F = k12_euclid_3(A,B,C,D,E)
<=> ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& k3_real_1(k3_real_1(G,H),F) = np__1
& E = k17_euclid(A,k17_euclid(A,k18_euclid(G,A,B),k18_euclid(H,A,C)),k18_euclid(F,A,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(d14_euclid_3,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
=> ( D = k13_euclid_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr),D,E) = k10_euclid_3(np__2,A,B,C,E) ) ) ) ) ) ) ).
fof(d15_euclid_3,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
=> ( D = k14_euclid_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr),D,E) = k11_euclid_3(np__2,A,B,C,E) ) ) ) ) ) ) ).
fof(d16_euclid_3,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
=> ( D = k15_euclid_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr),D,E) = k12_euclid_3(np__2,A,B,C,E) ) ) ) ) ) ) ).
fof(t64_euclid_3,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)))
=> ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
=> ( r2_hidden(D,k8_euclid_3(np__2,A,B,C))
<=> ~ ( r1_xreal_0(np__0,k10_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k11_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k12_euclid_3(np__2,A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t65_euclid_3,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)))
=> ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
=> ( r2_hidden(D,k5_euclid_3(np__2,A,B,C))
<=> ( r1_xreal_0(np__0,k10_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k11_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k12_euclid_3(np__2,A,B,C,D))
& ~ ( k10_euclid_3(np__2,A,B,C,D) != np__0
& k11_euclid_3(np__2,A,B,C,D) != np__0
& k12_euclid_3(np__2,A,B,C,D) != np__0 ) ) ) ) ) ) ) ) ).
fof(t66_euclid_3,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)))
=> ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
=> ( r2_hidden(D,k5_euclid_3(np__2,A,B,C))
<=> ~ ( ~ ( k10_euclid_3(np__2,A,B,C,D) = np__0
& r1_xreal_0(np__0,k11_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k12_euclid_3(np__2,A,B,C,D)) )
& ~ ( r1_xreal_0(np__0,k10_euclid_3(np__2,A,B,C,D))
& k11_euclid_3(np__2,A,B,C,D) = np__0
& r1_xreal_0(np__0,k12_euclid_3(np__2,A,B,C,D)) )
& ~ ( r1_xreal_0(np__0,k10_euclid_3(np__2,A,B,C,D))
& r1_xreal_0(np__0,k11_euclid_3(np__2,A,B,C,D))
& k12_euclid_3(np__2,A,B,C,D) = np__0 ) ) ) ) ) ) ) ) ).
fof(t67_euclid_3,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)))
=> ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
=> ( r2_hidden(D,k7_euclid_3(np__2,A,B,C))
<=> ( ~ r1_xreal_0(k10_euclid_3(np__2,A,B,C,D),np__0)
& ~ r1_xreal_0(k11_euclid_3(np__2,A,B,C,D),np__0)
& ~ r1_xreal_0(k12_euclid_3(np__2,A,B,C,D),np__0) ) ) ) ) ) ) ) ).
fof(t68_euclid_3,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)))
=> ~ ( r1_euclid_3(np__2,k20_euclid(np__2,B,A),k20_euclid(np__2,C,A))
& v1_xboole_0(k7_euclid_3(np__2,A,B,C)) ) ) ) ) ).
fof(dt_k1_euclid_3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k1_euclid_3(A),u1_struct_0(k15_euclid(np__2))) ) ).
fof(dt_k2_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(k2_euclid_3(A),k2_numbers) ) ).
fof(dt_k3_euclid_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(k3_euclid_3(A),k1_numbers) ) ).
fof(dt_k4_euclid_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> m1_subset_1(k4_euclid_3(A,B,C),k1_numbers) ) ).
fof(dt_k5_euclid_3,axiom,
! [A,B,C,D] :
( ( 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(D,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k5_euclid_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k6_euclid_3,axiom,
! [A,B,C,D] :
( ( 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(D,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k6_euclid_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k7_euclid_3,axiom,
! [A,B,C,D] :
( ( 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(D,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k7_euclid_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k8_euclid_3,axiom,
! [A,B,C,D] :
( ( 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(D,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k8_euclid_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k9_euclid_3,axiom,
! [A,B,C,D] :
( ( 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(D,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k9_euclid_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k10_euclid_3,axiom,
! [A,B,C,D,E] :
( ( 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(D,u1_struct_0(k15_euclid(A)))
& m1_subset_1(E,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k10_euclid_3(A,B,C,D,E),k1_numbers) ) ).
fof(dt_k11_euclid_3,axiom,
! [A,B,C,D,E] :
( ( 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(D,u1_struct_0(k15_euclid(A)))
& m1_subset_1(E,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k11_euclid_3(A,B,C,D,E),k1_numbers) ) ).
fof(dt_k12_euclid_3,axiom,
! [A,B,C,D,E] :
( ( 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(D,u1_struct_0(k15_euclid(A)))
& m1_subset_1(E,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k12_euclid_3(A,B,C,D,E),k1_numbers) ) ).
fof(dt_k13_euclid_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( v1_funct_1(k13_euclid_3(A,B,C))
& v1_funct_2(k13_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(k13_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) ) ) ).
fof(dt_k14_euclid_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( v1_funct_1(k14_euclid_3(A,B,C))
& v1_funct_2(k14_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(k14_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) ) ) ).
fof(dt_k15_euclid_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( v1_funct_1(k15_euclid_3(A,B,C))
& v1_funct_2(k15_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(k15_euclid_3(A,B,C),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) ) ) ).
fof(d6_euclid_3,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)))
=> k6_euclid_3(A,B,C,D) = a_4_0_euclid_3(A,B,C,D) ) ) ) ) ).
fof(d8_euclid_3,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)))
=> k8_euclid_3(A,B,C,D) = a_4_1_euclid_3(A,B,C,D) ) ) ) ) ).
fof(t62_euclid_3,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)))
=> k9_euclid_3(A,B,C,D) = a_4_2_euclid_3(A,B,C,D) ) ) ) ) ).
fof(fraenkel_a_4_0_euclid_3,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_0_euclid_3(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = F
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& ? [I] :
( m1_subset_1(I,k1_numbers)
& r1_xreal_0(np__0,G)
& r1_xreal_0(np__0,H)
& r1_xreal_0(np__0,I)
& k3_real_1(k3_real_1(G,H),I) = np__1
& F = k17_euclid(B,k17_euclid(B,k18_euclid(G,B,C),k18_euclid(H,B,D)),k18_euclid(I,B,E)) ) ) ) ) ) ) ).
fof(fraenkel_a_4_1_euclid_3,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_3(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = F
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& ? [I] :
( m1_subset_1(I,k1_numbers)
& ~ ( r1_xreal_0(np__0,G)
& r1_xreal_0(np__0,H)
& r1_xreal_0(np__0,I) )
& k3_real_1(k3_real_1(G,H),I) = np__1
& F = k17_euclid(B,k17_euclid(B,k18_euclid(G,B,C),k18_euclid(H,B,D)),k18_euclid(I,B,E)) ) ) ) ) ) ) ).
fof(fraenkel_a_4_2_euclid_3,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_2_euclid_3(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = F
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& ? [I] :
( m1_subset_1(I,k1_numbers)
& k3_real_1(k3_real_1(G,H),I) = np__1
& F = k17_euclid(B,k17_euclid(B,k18_euclid(G,B,C),k18_euclid(H,B,D)),k18_euclid(I,B,E)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------