SET007 Axioms: SET007+316.ax
%------------------------------------------------------------------------------
% File : SET007+316 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The 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 : euclid [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 117 ( 2 unt; 0 def)
% Number of atoms : 465 ( 115 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 359 ( 11 ~; 0 |; 78 &)
% ( 9 <=>; 261 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 59 ( 59 usr; 6 con; 0-5 aty)
% Number of variables : 281 ( 279 !; 2 ?)
% 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,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k14_euclid(A))
& v1_metric_1(k14_euclid(A))
& v6_metric_1(k14_euclid(A))
& v7_metric_1(k14_euclid(A))
& v8_metric_1(k14_euclid(A))
& v9_metric_1(k14_euclid(A)) ) ) ).
fof(fc2_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k15_euclid(A))
& v1_pre_topc(k15_euclid(A))
& v2_pre_topc(k15_euclid(A)) ) ) ).
fof(d1_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ).
fof(d2_euclid,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k1_numbers,k1_numbers)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k2_euclid
<=> ! [B] :
( v1_xreal_0(B)
=> k1_funct_1(A,B) = k18_complex1(B) ) ) ) ).
fof(d3_euclid,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k3_euclid(A) = k5_finseqop(k1_numbers,k1_numbers,A,k2_euclid) ) ).
fof(d4_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_euclid(A) = k4_finseqop(k1_numbers,A,np__0) ) ).
fof(d5_euclid,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k12_euclid(A) = k9_square_1(k15_rvsum_1(k11_rvsum_1(A))) ) ).
fof(t1_euclid,axiom,
$true ).
fof(t2_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k3_finseq_1(B) = A ) ) ).
fof(t3_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k4_finseq_1(B) = k2_finseq_1(A) ) ) ).
fof(t4_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> r2_hidden(k1_funct_1(C,B),k1_numbers) ) ) ) ).
fof(t5_euclid,axiom,
! [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_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) )
=> B = C ) ) ) ) ).
fof(t6_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( C = k1_funct_1(D,B)
=> k1_funct_1(k10_euclid(A,D),B) = k18_complex1(C) ) ) ) ) ) ).
fof(t7_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k10_euclid(A,k5_euclid(A)) = k4_finseqop(k1_numbers,A,np__0) ) ).
fof(t8_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k10_euclid(A,k6_euclid(A,B)) = k10_euclid(A,B) ) ) ).
fof(t9_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> k10_euclid(A,k9_euclid(A,B,C)) = k10_rvsum_1(A,k18_complex1(B),k10_euclid(A,C)) ) ) ) ).
fof(t10_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k12_euclid(k5_euclid(A)) = np__0 ) ).
fof(t11_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ( k12_euclid(B) = np__0
=> B = k5_euclid(A) ) ) ) ).
fof(t12_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> r1_xreal_0(np__0,k12_euclid(B)) ) ) ).
fof(t13_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k12_euclid(k6_euclid(A,B)) = k12_euclid(B) ) ) ).
fof(t14_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> k12_euclid(k9_euclid(A,B,C)) = k4_real_1(k18_complex1(B),k12_euclid(C)) ) ) ) ).
fof(t15_euclid,axiom,
! [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))
=> r1_xreal_0(k12_euclid(k7_euclid(A,B,C)),k3_real_1(k12_euclid(B),k12_euclid(C))) ) ) ) ).
fof(t16_euclid,axiom,
! [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))
=> r1_xreal_0(k12_euclid(k8_euclid(A,B,C)),k3_real_1(k12_euclid(B),k12_euclid(C))) ) ) ) ).
fof(t17_euclid,axiom,
! [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))
=> r1_xreal_0(k5_real_1(k12_euclid(B),k12_euclid(C)),k12_euclid(k7_euclid(A,B,C))) ) ) ) ).
fof(t18_euclid,axiom,
! [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))
=> r1_xreal_0(k5_real_1(k12_euclid(B),k12_euclid(C)),k12_euclid(k8_euclid(A,B,C))) ) ) ) ).
fof(t19_euclid,axiom,
! [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))
=> ( k12_euclid(k8_euclid(A,B,C)) = np__0
<=> B = C ) ) ) ) ).
fof(t20_euclid,axiom,
! [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))
=> ~ ( B != C
& r1_xreal_0(k12_euclid(k8_euclid(A,B,C)),np__0) ) ) ) ) ).
fof(t21_euclid,axiom,
! [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))
=> k12_euclid(k8_euclid(A,B,C)) = k12_euclid(k8_euclid(A,C,B)) ) ) ) ).
fof(t22_euclid,axiom,
! [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))
=> r1_xreal_0(k12_euclid(k8_euclid(A,B,C)),k3_real_1(k12_euclid(k8_euclid(A,B,D)),k12_euclid(k8_euclid(A,D,C)))) ) ) ) ) ).
fof(d6_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) )
=> ( B = k13_euclid(A)
<=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k1_metric_1(k1_euclid(A),k1_euclid(A),B,C,D) = k12_euclid(k8_euclid(A,C,D)) ) ) ) ) ) ).
fof(t23_euclid,axiom,
! [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))
=> k11_euclid(A,k8_euclid(A,B,C)) = k11_euclid(A,k8_euclid(A,C,B)) ) ) ) ).
fof(t24_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_pcomps_1(k1_euclid(A),k13_euclid(A)) ) ).
fof(d7_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ).
fof(d8_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ).
fof(t25_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> u1_struct_0(k15_euclid(A)) = k1_euclid(A) ) ).
fof(t26_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k1_numbers)
& m2_relset_1(B,k2_finseq_1(A),k1_numbers) ) ) ) ).
fof(t27_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> m2_finseq_1(B,k1_numbers) ) ) ).
fof(t28_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = B
=> k3_finseq_1(C) = A ) ) ) ) ).
fof(d9_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k16_euclid(A) = k5_euclid(A) ) ).
fof(d10_euclid,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)))
=> ( D = k17_euclid(A,B,C)
<=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ( ( E = B
& F = C )
=> D = k7_euclid(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t29_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k12_rvsum_1(A,k10_euclid(A,B)) = k11_euclid(A,B) ) ) ).
fof(t30_euclid,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)))
=> k17_euclid(A,k17_euclid(A,B,C),D) = k17_euclid(A,B,k17_euclid(A,C,D)) ) ) ) ) ).
fof(t31_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ( k17_euclid(A,k16_euclid(A),B) = B
& k17_euclid(A,B,k16_euclid(A)) = B ) ) ) ).
fof(d11_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
=> ( D = k18_euclid(A,B,C)
<=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
=> ( E = C
=> D = k9_euclid(B,A,E) ) ) ) ) ) ) ) ).
fof(t32_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> k18_euclid(B,A,k16_euclid(A)) = k16_euclid(A) ) ) ).
fof(t33_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ( k18_euclid(np__1,A,B) = B
& k18_euclid(np__0,A,B) = k16_euclid(A) ) ) ) ).
fof(t34_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k18_euclid(k3_xcmplx_0(C,D),A,B) = k18_euclid(C,A,k18_euclid(D,A,B)) ) ) ) ) ).
fof(t35_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( k18_euclid(C,A,B) = k16_euclid(A)
& C != np__0
& B != k16_euclid(A) ) ) ) ) ).
fof(t36_euclid,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_xreal_0(D)
=> k18_euclid(D,A,k17_euclid(A,B,C)) = k17_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ).
fof(t37_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k18_euclid(k2_xcmplx_0(C,D),A,B) = k17_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ).
fof(t38_euclid,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_xreal_0(D)
=> ~ ( k18_euclid(D,A,B) = k18_euclid(D,A,C)
& D != np__0
& B != C ) ) ) ) ) ).
fof(d12_euclid,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)))
=> ( C = k19_euclid(A,B)
<=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( D = B
=> C = k6_euclid(A,D) ) ) ) ) ) ) ).
fof(t39_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k19_euclid(A,k19_euclid(A,B)) = B ) ) ).
fof(t40_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k17_euclid(A,B,k19_euclid(A,B)) = k16_euclid(A) ) ) ).
fof(t41_euclid,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)))
=> ( k17_euclid(A,B,C) = k16_euclid(A)
=> ( B = k19_euclid(A,C)
& C = k19_euclid(A,B) ) ) ) ) ) ).
fof(t42_euclid,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)))
=> k19_euclid(A,k17_euclid(A,B,C)) = k17_euclid(A,k19_euclid(A,B),k19_euclid(A,C)) ) ) ) ).
fof(t43_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k19_euclid(A,B) = k18_euclid(k1_real_1(np__1),A,B) ) ) ).
fof(t44_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> ( k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(k4_xcmplx_0(C),A,B)
& k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(C,A,k19_euclid(A,B)) ) ) ) ) ).
fof(d13_euclid,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)))
=> ( D = k20_euclid(A,B,C)
<=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ( ( E = B
& F = C )
=> D = k8_euclid(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t45_euclid,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)))
=> k20_euclid(A,B,C) = k17_euclid(A,B,k19_euclid(A,C)) ) ) ) ).
fof(t46_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k20_euclid(A,B,B) = k16_euclid(A) ) ) ).
fof(t47_euclid,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)))
=> ( k20_euclid(A,B,C) = k16_euclid(A)
=> B = C ) ) ) ) ).
fof(t48_euclid,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)))
=> ( k19_euclid(A,k20_euclid(A,B,C)) = k20_euclid(A,C,B)
& k19_euclid(A,k20_euclid(A,B,C)) = k17_euclid(A,k19_euclid(A,B),C) ) ) ) ) ).
fof(t49_euclid,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)))
=> k17_euclid(A,B,k20_euclid(A,C,D)) = k20_euclid(A,k17_euclid(A,B,C),D) ) ) ) ) ).
fof(t50_euclid,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)))
=> k20_euclid(A,B,k17_euclid(A,C,D)) = k20_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ).
fof(t51_euclid,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)))
=> k20_euclid(A,B,k20_euclid(A,C,D)) = k17_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ).
fof(t52_euclid,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)))
=> ( B = k20_euclid(A,k17_euclid(A,B,C),C)
& B = k17_euclid(A,k20_euclid(A,B,C),C) ) ) ) ) ).
fof(t53_euclid,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_xreal_0(D)
=> k18_euclid(D,A,k20_euclid(A,B,C)) = k20_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ).
fof(t54_euclid,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k18_euclid(k6_xcmplx_0(C,D),A,B) = k20_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ).
fof(t55_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& A = k10_finseq_1(B,C) ) ) ) ).
fof(d14_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k21_euclid(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = C
=> B = k1_funct_1(C,np__1) ) ) ) ) ) ).
fof(d15_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k22_euclid(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = C
=> B = k1_funct_1(C,np__2) ) ) ) ) ) ).
fof(d16_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k23_euclid(A,B) = k10_finseq_1(A,B) ) ) ).
fof(t56_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k21_euclid(k23_euclid(A,B)) = A
& k22_euclid(k23_euclid(A,B)) = B ) ) ) ).
fof(t57_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> A = k23_euclid(k21_euclid(A),k22_euclid(A)) ) ).
fof(t58_euclid,axiom,
k16_euclid(np__2) = k23_euclid(np__0,np__0) ).
fof(t59_euclid,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)))
=> k17_euclid(np__2,A,B) = k23_euclid(k3_real_1(k21_euclid(A),k21_euclid(B)),k3_real_1(k22_euclid(A),k22_euclid(B))) ) ) ).
fof(t60_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k17_euclid(np__2,k23_euclid(A,B),k23_euclid(C,D)) = k23_euclid(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ).
fof(t61_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k18_euclid(A,np__2,B) = k23_euclid(k3_xcmplx_0(A,k21_euclid(B)),k3_xcmplx_0(A,k22_euclid(B))) ) ) ).
fof(t62_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k18_euclid(A,np__2,k23_euclid(B,C)) = k23_euclid(k3_xcmplx_0(A,B),k3_xcmplx_0(A,C)) ) ) ) ).
fof(t63_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k19_euclid(np__2,A) = k23_euclid(k1_real_1(k21_euclid(A)),k1_real_1(k22_euclid(A))) ) ).
fof(t64_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k19_euclid(np__2,k23_euclid(A,B)) = k23_euclid(k4_xcmplx_0(A),k4_xcmplx_0(B)) ) ) ).
fof(t65_euclid,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)))
=> k20_euclid(np__2,A,B) = k23_euclid(k5_real_1(k21_euclid(A),k21_euclid(B)),k5_real_1(k22_euclid(A),k22_euclid(B))) ) ) ).
fof(t66_euclid,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k20_euclid(np__2,k23_euclid(A,B),k23_euclid(C,D)) = k23_euclid(k6_xcmplx_0(A,C),k6_xcmplx_0(B,D)) ) ) ) ) ).
fof(dt_k1_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k1_euclid(A))
& m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ).
fof(dt_k2_euclid,axiom,
( v1_funct_1(k2_euclid)
& v1_funct_2(k2_euclid,k1_numbers,k1_numbers)
& m2_relset_1(k2_euclid,k1_numbers,k1_numbers) ) ).
fof(dt_k3_euclid,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m2_finseq_1(k3_euclid(A),k1_numbers) ) ).
fof(dt_k4_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_finseq_1(k4_euclid(A),k1_numbers) ) ).
fof(dt_k5_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ).
fof(redefinition_k5_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k5_euclid(A) = k4_euclid(A) ) ).
fof(dt_k6_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> m2_finseq_2(k6_euclid(A,B),k1_numbers,k1_euclid(A)) ) ).
fof(involutiveness_k6_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> k6_euclid(A,k6_euclid(A,B)) = B ) ).
fof(redefinition_k6_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> k6_euclid(A,B) = k5_rvsum_1(B) ) ).
fof(dt_k7_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> m2_finseq_2(k7_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ).
fof(commutativity_k7_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k7_euclid(A,B,C) = k7_euclid(A,C,B) ) ).
fof(redefinition_k7_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k7_euclid(A,B,C) = k3_rvsum_1(B,C) ) ).
fof(dt_k8_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> m2_finseq_2(k8_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ).
fof(redefinition_k8_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> k8_euclid(A,B,C) = k7_rvsum_1(B,C) ) ).
fof(dt_k9_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_xreal_0(B)
& m1_subset_1(C,k1_euclid(A)) )
=> m2_finseq_2(k9_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ).
fof(redefinition_k9_euclid,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_xreal_0(B)
& m1_subset_1(C,k1_euclid(A)) )
=> k9_euclid(A,B,C) = k9_rvsum_1(B,C) ) ).
fof(dt_k10_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> m2_finseq_2(k10_euclid(A,B),k1_numbers,k4_finseq_2(A,k1_numbers)) ) ).
fof(redefinition_k10_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> k10_euclid(A,B) = k3_euclid(B) ) ).
fof(dt_k11_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> m2_finseq_2(k11_euclid(A,B),k1_numbers,k4_finseq_2(A,k1_numbers)) ) ).
fof(redefinition_k11_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A)) )
=> k11_euclid(A,B) = k11_rvsum_1(B) ) ).
fof(dt_k12_euclid,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m1_subset_1(k12_euclid(A),k1_numbers) ) ).
fof(dt_k13_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k13_euclid(A))
& v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers)
& m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ).
fof(dt_k14_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_metric_1(k14_euclid(A))
& v6_metric_1(k14_euclid(A))
& v7_metric_1(k14_euclid(A))
& v8_metric_1(k14_euclid(A))
& v9_metric_1(k14_euclid(A))
& l1_metric_1(k14_euclid(A)) ) ) ).
fof(dt_k15_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_pre_topc(k15_euclid(A))
& v2_pre_topc(k15_euclid(A))
& l1_pre_topc(k15_euclid(A)) ) ) ).
fof(dt_k16_euclid,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k16_euclid(A),u1_struct_0(k15_euclid(A))) ) ).
fof(dt_k17_euclid,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(k17_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ).
fof(commutativity_k17_euclid,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))) )
=> k17_euclid(A,B,C) = k17_euclid(A,C,B) ) ).
fof(dt_k18_euclid,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B))) )
=> m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ).
fof(dt_k19_euclid,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k19_euclid(A,B),u1_struct_0(k15_euclid(A))) ) ).
fof(dt_k20_euclid,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(k20_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ).
fof(dt_k21_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(k21_euclid(A),k1_numbers) ) ).
fof(dt_k22_euclid,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> m1_subset_1(k22_euclid(A),k1_numbers) ) ).
fof(dt_k23_euclid,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k23_euclid(A,B),u1_struct_0(k15_euclid(np__2))) ) ).
%------------------------------------------------------------------------------