SET007 Axioms: SET007+402.ax
%------------------------------------------------------------------------------
% File : SET007+402 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Sequences in E^N_T
% 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 : toprns_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 69 ( 5 unt; 0 def)
% Number of atoms : 479 ( 49 equ)
% Maximal formula atoms : 13 ( 6 avg)
% Number of connectives : 429 ( 19 ~; 1 |; 196 &)
% ( 14 <=>; 199 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 36 ( 36 usr; 6 con; 0-4 aty)
% Number of variables : 203 ( 195 !; 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(t1_toprns_1,axiom,
$true ).
fof(t2_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
<=> ( k1_relat_1(B) = k5_numbers
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> m1_subset_1(k1_funct_1(B,C),u1_struct_0(k15_euclid(A))) ) ) ) ) ) ).
fof(d1_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_toprns_1(B,A)
<=> r1_tarski(k2_relat_1(B),k4_xboole_0(u1_struct_0(k15_euclid(A)),k1_struct_0(k15_euclid(A),k16_euclid(A)))) ) ) ) ).
fof(t3_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_toprns_1(B,A)
<=> ! [C] :
~ ( r2_hidden(C,k5_numbers)
& k1_funct_1(B,C) = k16_euclid(A) ) ) ) ) ).
fof(t4_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_toprns_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(A),B,C) != k16_euclid(A) ) ) ) ) ).
fof(t5_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ! [D] :
( r2_hidden(D,k5_numbers)
=> k1_funct_1(B,D) = k1_funct_1(C,D) )
=> B = C ) ) ) ) ).
fof(t6_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(A),B,D) = k2_normsp_1(k15_euclid(A),C,D) )
=> B = C ) ) ) ) ).
fof(d2_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( D = k1_toprns_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(A),D,E) = k17_euclid(A,k2_normsp_1(k15_euclid(A),B,E),k2_normsp_1(k15_euclid(A),C,E)) ) ) ) ) ) ) ).
fof(d3_toprns_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(B)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(B))) )
=> ( D = k2_toprns_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(B),D,E) = k18_euclid(A,B,k2_normsp_1(k15_euclid(B),C,E)) ) ) ) ) ) ) ).
fof(d4_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( C = k3_toprns_1(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(A),C,D) = k19_euclid(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) ).
fof(d5_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k4_toprns_1(A,B,C) = k1_toprns_1(A,B,k3_toprns_1(A,C)) ) ) ) ).
fof(d6_toprns_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_numbers)
=> ( C = k5_toprns_1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k1_numbers)
& B = D
& C = k12_euclid(D) ) ) ) ) ) ).
fof(d7_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k6_toprns_1(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) ).
fof(t7_toprns_1,axiom,
$true ).
fof(t8_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) ).
fof(t9_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k6_toprns_1(A,k2_toprns_1(B,A,C)) = k14_seq_1(k6_toprns_1(A,C),k18_complex1(B)) ) ) ) ).
fof(t10_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k1_toprns_1(A,B,C) = k1_toprns_1(A,C,B) ) ) ) ).
fof(t11_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k1_toprns_1(A,k1_toprns_1(A,B,C),D) = k1_toprns_1(A,B,k1_toprns_1(A,C,D)) ) ) ) ) ).
fof(t12_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k3_toprns_1(A,B) = k2_toprns_1(k1_real_1(np__1),A,B) ) ) ).
fof(t13_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k2_toprns_1(B,A,k1_toprns_1(A,C,D)) = k1_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) ).
fof(t14_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k2_toprns_1(k4_real_1(B,C),A,D) = k2_toprns_1(B,A,k2_toprns_1(C,A,D)) ) ) ) ) ).
fof(t15_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k2_toprns_1(B,A,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) ).
fof(t16_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k4_toprns_1(A,B,k1_toprns_1(A,C,D)) = k4_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) ).
fof(t17_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k2_toprns_1(np__1,A,B) = B ) ) ).
fof(t18_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k3_toprns_1(A,k3_toprns_1(A,B)) = B ) ) ).
fof(t19_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k4_toprns_1(A,B,k3_toprns_1(A,C)) = k1_toprns_1(A,B,C) ) ) ) ).
fof(t20_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k4_toprns_1(A,B,k4_toprns_1(A,C,D)) = k1_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) ).
fof(t21_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> k1_toprns_1(A,B,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k1_toprns_1(A,B,C),D) ) ) ) ) ).
fof(t22_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_toprns_1(C,A)
=> ( B = np__0
| v1_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) ) ).
fof(t23_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_toprns_1(B,A)
=> v1_toprns_1(k3_toprns_1(A,B),A) ) ) ) ).
fof(t24_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_toprns_1(A,k16_euclid(A)) = np__0 ) ).
fof(t25_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ( k5_toprns_1(A,B) = np__0
=> B = k16_euclid(A) ) ) ) ).
fof(t26_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> r1_xreal_0(np__0,k5_toprns_1(A,B)) ) ) ).
fof(t27_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k5_toprns_1(A,k19_euclid(A,B)) = k5_toprns_1(A,B) ) ) ).
fof(t28_toprns_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)))
=> k5_toprns_1(A,k20_euclid(A,B,C)) = k5_toprns_1(A,k20_euclid(A,C,B)) ) ) ) ).
fof(t29_toprns_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)))
=> ( k5_toprns_1(A,k20_euclid(A,B,C)) = np__0
<=> B = C ) ) ) ) ).
fof(t30_toprns_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_xreal_0(k5_toprns_1(A,k17_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) ).
fof(t31_toprns_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_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) ).
fof(t32_toprns_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_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k17_euclid(A,B,C))) ) ) ) ).
fof(t33_toprns_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_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ) ).
fof(t34_toprns_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)))
=> ~ ( B != C
& r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),np__0) ) ) ) ) ).
fof(t35_toprns_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_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,k20_euclid(A,B,D)),k5_toprns_1(A,k20_euclid(A,D,C)))) ) ) ) ) ).
fof(t36_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( r1_xreal_0(np__0,B)
& ~ r1_xreal_0(k5_toprns_1(A,E),k5_toprns_1(A,D))
& ~ r1_xreal_0(C,B)
& r1_xreal_0(k4_real_1(k5_toprns_1(A,E),C),k4_real_1(k5_toprns_1(A,D),B)) ) ) ) ) ) ) ).
fof(t37_toprns_1,axiom,
$true ).
fof(t38_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( ( ~ r1_xreal_0(B,k1_real_1(k5_toprns_1(A,C)))
& ~ r1_xreal_0(k5_toprns_1(A,C),B) )
<=> ~ r1_xreal_0(k5_toprns_1(A,C),k18_complex1(B)) ) ) ) ) ).
fof(d8_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v2_toprns_1(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(C,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D))) ) ) ) ) ) ).
fof(t39_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(E,C)
& r1_xreal_0(D,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,E))) ) ) ) ) ) ) ).
fof(d9_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) ).
fof(d10_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( C = k7_toprns_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) ) ).
fof(t40_toprns_1,axiom,
$true ).
fof(t41_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ( v3_toprns_1(B,A)
& v3_toprns_1(C,A) )
=> v3_toprns_1(k1_toprns_1(A,B,C),A) ) ) ) ) ).
fof(t42_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ( v3_toprns_1(B,A)
& v3_toprns_1(C,A) )
=> k7_toprns_1(A,k1_toprns_1(A,B,C)) = k17_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) ).
fof(t43_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(C,A)
=> v3_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) ).
fof(t44_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(C,A)
=> k7_toprns_1(A,k2_toprns_1(B,A,C)) = k18_euclid(B,A,k7_toprns_1(A,C)) ) ) ) ) ).
fof(t45_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(B,A)
=> v3_toprns_1(k3_toprns_1(A,B),A) ) ) ) ).
fof(t46_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(B,A)
=> k7_toprns_1(A,k3_toprns_1(A,B)) = k19_euclid(A,k7_toprns_1(A,B)) ) ) ) ).
fof(t47_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ( v3_toprns_1(B,A)
& v3_toprns_1(C,A) )
=> v3_toprns_1(k4_toprns_1(A,B,C),A) ) ) ) ) ).
fof(t48_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( ( v3_toprns_1(B,A)
& v3_toprns_1(C,A) )
=> k7_toprns_1(A,k4_toprns_1(A,B,C)) = k20_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) ).
fof(t49_toprns_1,axiom,
$true ).
fof(t50_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v3_toprns_1(B,A)
=> v2_toprns_1(B,A) ) ) ) ).
fof(t51_toprns_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ~ ( v3_toprns_1(B,A)
& k7_toprns_1(A,B) != k16_euclid(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& r1_xreal_0(k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)),k6_real_1(k5_toprns_1(A,k7_toprns_1(A,B)),np__2)) ) ) ) ) ) ).
fof(s1_toprns_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
& m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_normsp_1(k15_euclid(f1_s1_toprns_1),A,B) = f2_s1_toprns_1(B) ) ) ).
fof(dt_k1_toprns_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_funct_1(k1_toprns_1(A,B,C))
& v1_funct_2(k1_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(k1_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) ).
fof(dt_k2_toprns_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k5_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
& m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
=> ( v1_funct_1(k2_toprns_1(A,B,C))
& v1_funct_2(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B)))
& m2_relset_1(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B))) ) ) ).
fof(dt_k3_toprns_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_funct_1(k3_toprns_1(A,B))
& v1_funct_2(k3_toprns_1(A,B),k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(k3_toprns_1(A,B),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) ).
fof(dt_k4_toprns_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_funct_1(k4_toprns_1(A,B,C))
& v1_funct_2(k4_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A)))
& m2_relset_1(k4_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) ).
fof(dt_k5_toprns_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ).
fof(dt_k6_toprns_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> ( v1_funct_1(k6_toprns_1(A,B))
& v1_funct_2(k6_toprns_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k6_toprns_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k7_toprns_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
& m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k7_toprns_1(A,B),u1_struct_0(k15_euclid(A))) ) ).
%------------------------------------------------------------------------------