SET007 Axioms: SET007+839.ax
%------------------------------------------------------------------------------
% File : SET007+839 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Intersections of Intervals and Balls 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 : topreal9 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 97 ( 0 unt; 0 def)
% Number of atoms : 619 ( 99 equ)
% Maximal formula atoms : 23 ( 6 avg)
% Number of connectives : 572 ( 50 ~; 6 |; 169 &)
% ( 13 <=>; 334 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 10 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 63 ( 63 usr; 5 con; 0-4 aty)
% Number of variables : 366 ( 360 !; 6 ?)
% 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_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& ~ v2_xreal_0(C) )
=> v1_xboole_0(k1_topreal9(A,B,C)) ) ).
fof(fc2_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ v1_xboole_0(k1_topreal9(A,B,C)) ) ).
fof(fc3_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& v3_xreal_0(C) )
=> v1_xboole_0(k2_topreal9(A,B,C)) ) ).
fof(fc4_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ~ v1_xboole_0(k2_topreal9(A,B,C)) ) ).
fof(fc5_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v3_pre_topc(k1_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k1_topreal9(A,B,C),A) ) ) ).
fof(fc6_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v4_pre_topc(k2_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k2_topreal9(A,B,C),A) ) ) ).
fof(fc7_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ).
fof(fc8_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v3_pre_topc(k1_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k1_topreal9(A,B,C),A)
& v1_jordan1(k1_topreal9(A,B,C),A) ) ) ).
fof(fc9_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v4_pre_topc(k2_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k2_topreal9(A,B,C),A)
& v1_jordan1(k2_topreal9(A,B,C),A) ) ) ).
fof(fc10_topreal9,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k3_borsuk_1(k15_euclid(A),k15_euclid(A),k16_euclid(A)))
& v1_funct_2(k3_borsuk_1(k15_euclid(A),k15_euclid(A),k16_euclid(A)),u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& v1_topreal9(k3_borsuk_1(k15_euclid(A),k15_euclid(A),k16_euclid(A)),A)
& v2_topreal9(k3_borsuk_1(k15_euclid(A),k15_euclid(A),k16_euclid(A)),A) ) ) ).
fof(rc1_topreal9,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] :
( m1_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& v5_pre_topc(B,k15_euclid(A),k15_euclid(A))
& v1_topreal9(B,A)
& v2_topreal9(B,A) ) ) ).
fof(fc11_topreal9,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_funct_1(k2_jgraph_2(A,np__0,B,np__0))
& v1_funct_2(k2_jgraph_2(A,np__0,B,np__0),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)))
& v1_topreal9(k2_jgraph_2(A,np__0,B,np__0),np__2)
& v2_topreal9(k2_jgraph_2(A,np__0,B,np__0),np__2) ) ) ).
fof(fc12_topreal9,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(k4_topreal9(A,B,C)) ) ).
fof(fc13_topreal9,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(k4_topreal9(A,B,C))
& v1_jordan1(k4_topreal9(A,B,C),A) ) ) ).
fof(fc14_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& v3_xreal_0(C) )
=> ( v1_xboole_0(k3_topreal9(A,B,C))
& v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ).
fof(fc15_topreal9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ( ~ v1_xboole_0(k3_topreal9(A,B,C))
& v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A))
& v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ).
fof(fc16_topreal9,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ v1_xboole_0(k6_jgraph_6(A,B,C)) ) ).
fof(fc17_topreal9,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ~ v1_xboole_0(k7_jgraph_6(A,B,C)) ) ).
fof(fc18_topreal9,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C) )
=> v1_jordan1(k6_jgraph_6(A,B,C),np__2) ) ).
fof(fc19_topreal9,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C) )
=> v1_jordan1(k7_jgraph_6(A,B,C),np__2) ) ).
fof(t1_topreal9,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,k20_euclid(A,B,C),D) = k20_euclid(A,k20_euclid(A,B,D),C) ) ) ) ) ).
fof(t2_topreal9,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,C) = k17_euclid(A,B,D)
=> C = D ) ) ) ) ) ).
fof(t3_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ~ ( ~ v1_xboole_0(A)
& B = k17_euclid(A,B,k5_jordan2c(A)) ) ) ) ).
fof(t4_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( E = k17_euclid(A,k18_euclid(k6_xcmplx_0(np__1,B),A,C),k18_euclid(B,A,D))
=> ( ~ ( E = C
& B != np__0
& C != D )
& ( ( B = np__0
| C = D )
=> E = C )
& ~ ( E = D
& B != np__1
& C != D )
& ( ( B = np__1
| C = D )
=> E = D ) ) ) ) ) ) ) ).
fof(t5_topreal9,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k7_square_1(k12_euclid(A)) = k15_rvsum_1(k11_rvsum_1(A)) ) ).
fof(t6_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ~ ( C != D
& r2_hidden(C,k10_metric_1(B,E,A))
& r2_hidden(D,k10_metric_1(B,E,A))
& r1_xreal_0(A,np__0) ) ) ) ) ) ) ).
fof(t7_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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,k1_topreal9(A,D,B))
<=> ~ r1_xreal_0(B,k5_toprns_1(A,k20_euclid(A,C,D))) ) ) ) ) ) ).
fof(t8_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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,k2_topreal9(A,D,B))
<=> r1_xreal_0(k5_toprns_1(A,k20_euclid(A,C,D)),B) ) ) ) ) ) ).
fof(t9_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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,k3_topreal9(A,D,B))
<=> k5_toprns_1(A,k20_euclid(A,C,D)) = B ) ) ) ) ) ).
fof(t10_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ~ ( r2_hidden(C,k1_topreal9(A,k16_euclid(A),B))
& r1_xreal_0(B,k5_toprns_1(A,C)) ) ) ) ) ).
fof(t11_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(C,k2_topreal9(A,k16_euclid(A),B))
=> r1_xreal_0(k5_toprns_1(A,C),B) ) ) ) ) ).
fof(t12_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(C,k3_topreal9(A,k16_euclid(A),B))
=> k5_toprns_1(A,C) = B ) ) ) ) ).
fof(t13_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(A)))
=> ( C = D
=> k9_metric_1(k14_euclid(A),D,B) = k1_topreal9(A,C,B) ) ) ) ) ) ).
fof(t14_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(A)))
=> ( C = D
=> k10_metric_1(k14_euclid(A),D,B) = k2_topreal9(A,C,B) ) ) ) ) ) ).
fof(t15_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(A)))
=> ( C = D
=> k11_metric_1(k14_euclid(A),D,B) = k3_topreal9(A,C,B) ) ) ) ) ) ).
fof(t16_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_tarski(k1_topreal9(A,C,B),k2_topreal9(A,C,B)) ) ) ) ).
fof(t17_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_tarski(k3_topreal9(A,C,B),k2_topreal9(A,C,B)) ) ) ) ).
fof(t18_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k4_subset_1(u1_struct_0(k15_euclid(A)),k1_topreal9(A,C,B),k3_topreal9(A,C,B)) = k2_topreal9(A,C,B) ) ) ) ).
fof(t19_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_xboole_0(k1_topreal9(A,C,B),k3_topreal9(A,C,B)) ) ) ) ).
fof(t20_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ~ ( ~ v1_xboole_0(k1_topreal9(A,C,B))
& r1_xreal_0(B,np__0) ) ) ) ) ).
fof(t21_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( v1_xboole_0(k1_topreal9(A,C,B))
=> r1_xreal_0(B,np__0) ) ) ) ) ).
fof(t22_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( ~ v1_xboole_0(k2_topreal9(A,C,B))
=> r1_xreal_0(np__0,B) ) ) ) ) ).
fof(t23_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ~ ( v1_xboole_0(k2_topreal9(A,C,B))
& r1_xreal_0(np__0,B) ) ) ) ) ).
fof(t24_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(A)))
=> ( ( k2_xcmplx_0(B,C) = np__1
& k3_real_1(k18_complex1(B),k18_complex1(C)) = np__1
& r2_hidden(E,k2_topreal9(A,F,D))
& r2_hidden(G,k1_topreal9(A,F,D)) )
=> ( C = np__0
| r2_hidden(k17_euclid(A,k18_euclid(B,A,E),k18_euclid(C,A,G)),k1_topreal9(A,F,D)) ) ) ) ) ) ) ) ) ) ).
fof(d4_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A))) )
=> ( v1_topreal9(B,A)
<=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)),B,k18_euclid(C,A,D)) = k18_euclid(C,A,k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)),B,D)) ) ) ) ) ) ).
fof(d5_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A))) )
=> ( v2_topreal9(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)),B,k17_euclid(A,C,D)) = k17_euclid(A,k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)),B,C),k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)),B,D)) ) ) ) ) ) ).
fof(t25_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A)))
& v1_topreal9(B,A)
& v2_topreal9(B,A)
& m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_jordan1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> v1_jordan1(k4_pre_topc(k15_euclid(A),k15_euclid(A),B,C),A) ) ) ) ).
fof(t26_topreal9,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] :
( r2_hidden(D,k4_topreal9(A,B,C))
<=> ? [E] :
( v1_xreal_0(E)
& D = k17_euclid(A,k18_euclid(k6_xcmplx_0(np__1,E),A,B),k18_euclid(E,A,C))
& r1_xreal_0(np__0,E) ) ) ) ) ) ).
fof(t27_topreal9,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,k4_topreal9(A,B,C)) ) ) ) ).
fof(t28_topreal9,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,k4_topreal9(A,C,B)) ) ) ) ).
fof(t29_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k4_topreal9(A,B,B) = k1_struct_0(k15_euclid(A),B) ) ) ).
fof(t30_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(B,k4_topreal9(A,C,D))
=> r1_tarski(k4_topreal9(A,C,B),k4_topreal9(A,C,D)) ) ) ) ) ) ).
fof(t31_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(B,k4_topreal9(A,C,D))
=> ( B = C
| k4_topreal9(A,C,D) = k4_topreal9(A,C,B) ) ) ) ) ) ) ).
fof(t32_topreal9,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_tarski(k3_topreal1(A,B,C),k4_topreal9(A,B,C)) ) ) ) ).
fof(t33_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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(C,k3_topreal9(A,D,B))
& r2_hidden(E,k1_topreal9(A,D,B)) )
=> k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,E),k3_topreal9(A,D,B)) = k1_struct_0(k15_euclid(A),C) ) ) ) ) ) ) ).
fof(t34_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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(C,k3_topreal9(A,D,B))
& r2_hidden(E,k3_topreal9(A,D,B)) )
=> r1_tarski(k6_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,E),k2_struct_0(k15_euclid(A),C,E)),k1_topreal9(A,D,B)) ) ) ) ) ) ) ).
fof(t35_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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(C,k3_topreal9(A,D,B))
& r2_hidden(E,k3_topreal9(A,D,B)) )
=> k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,E),k3_topreal9(A,D,B)) = k2_struct_0(k15_euclid(A),C,E) ) ) ) ) ) ) ).
fof(t36_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [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(C,k3_topreal9(A,D,B))
& r2_hidden(E,k3_topreal9(A,D,B)) )
=> k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,E),k3_topreal9(A,D,B)) = k2_struct_0(k15_euclid(A),C,E) ) ) ) ) ) ) ).
fof(t37_topreal9,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,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(A))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(A))
=> ~ ( G = D
& H = E
& I = F
& D != E
& r2_hidden(D,k1_topreal9(A,F,B))
& C = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k3_xcmplx_0(np__2,k2_euclid_2(A,k20_euclid(A,E,D),k20_euclid(A,D,F)))),k8_square_1(k1_quin_1(k15_rvsum_1(k11_euclid(A,k8_euclid(A,H,G))),k3_xcmplx_0(np__2,k2_euclid_2(A,k20_euclid(A,E,D),k20_euclid(A,D,F))),k6_xcmplx_0(k15_rvsum_1(k11_euclid(A,k8_euclid(A,G,I))),k5_square_1(B))))),k4_real_1(np__2,k15_rvsum_1(k11_euclid(A,k8_euclid(A,H,G)))))
& ! [J] :
( m1_subset_1(J,u1_struct_0(k15_euclid(A)))
=> ~ ( k1_struct_0(k15_euclid(A),J) = k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,D,E),k3_topreal9(A,F,B))
& J = k17_euclid(A,k18_euclid(k6_xcmplx_0(np__1,C),A,D),k18_euclid(C,A,E)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t38_topreal9,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,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(A))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(A))
=> ~ ( G = k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E))
& H = E
& I = F
& D != E
& r2_hidden(D,k3_topreal9(A,F,B))
& r2_hidden(E,k2_topreal9(A,F,B))
& ! [J] :
( m1_subset_1(J,u1_struct_0(k15_euclid(A)))
=> ~ ( J != D
& k2_struct_0(k15_euclid(A),D,J) = k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,D,E),k3_topreal9(A,F,B))
& ( r2_hidden(E,k3_topreal9(A,F,B))
=> J = E )
& ( C = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k3_xcmplx_0(np__2,k2_euclid_2(A,k20_euclid(A,E,k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E))),k20_euclid(A,k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E)),F)))),k8_square_1(k1_quin_1(k15_rvsum_1(k11_euclid(A,k8_euclid(A,H,G))),k3_xcmplx_0(np__2,k2_euclid_2(A,k20_euclid(A,E,k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E))),k20_euclid(A,k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E)),F))),k6_xcmplx_0(k15_rvsum_1(k11_euclid(A,k8_euclid(A,G,I))),k5_square_1(B))))),k4_real_1(np__2,k15_rvsum_1(k11_euclid(A,k8_euclid(A,H,G)))))
=> ( r2_hidden(E,k3_topreal9(A,F,B))
| J = k17_euclid(A,k18_euclid(k6_xcmplx_0(np__1,C),A,k17_euclid(A,k18_euclid(k6_real_1(np__1,np__2),A,D),k18_euclid(k6_real_1(np__1,np__2),A,E))),k18_euclid(C,A,E)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t39_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( ~ v1_xboole_0(k3_topreal9(A,C,B))
=> r1_xreal_0(np__0,B) ) ) ) ) ).
fof(t40_topreal9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ~ ( ~ v1_xboole_0(A)
& v1_xboole_0(k3_topreal9(A,C,B))
& r1_xreal_0(np__0,B) ) ) ) ) ).
fof(t41_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> k21_euclid(k17_euclid(np__2,k18_euclid(A,np__2,C),k18_euclid(B,np__2,D))) = k2_xcmplx_0(k3_xcmplx_0(A,k21_euclid(C)),k3_xcmplx_0(B,k21_euclid(D))) ) ) ) ) ).
fof(t42_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> k22_euclid(k17_euclid(np__2,k18_euclid(A,np__2,C),k18_euclid(B,np__2,D))) = k2_xcmplx_0(k3_xcmplx_0(A,k22_euclid(C)),k3_xcmplx_0(B,k22_euclid(D))) ) ) ) ) ).
fof(t43_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(D,k5_jgraph_6(A,B,C))
<=> k5_toprns_1(np__2,k20_euclid(np__2,D,k23_euclid(A,B))) = C ) ) ) ) ) ).
fof(t44_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(D,k7_jgraph_6(A,B,C))
<=> r1_xreal_0(k5_toprns_1(np__2,k20_euclid(np__2,D,k23_euclid(A,B))),C) ) ) ) ) ) ).
fof(t45_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(D,k6_jgraph_6(A,B,C))
<=> ~ r1_xreal_0(C,k5_toprns_1(np__2,k20_euclid(np__2,D,k23_euclid(A,B)))) ) ) ) ) ) ).
fof(t46_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> r1_tarski(k5_jgraph_6(A,B,C),k7_jgraph_6(A,B,C)) ) ) ) ).
fof(t47_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ( D = k23_euclid(A,B)
=> k10_metric_1(k14_euclid(np__2),D,C) = k7_jgraph_6(A,B,C) ) ) ) ) ) ).
fof(t48_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ( D = k23_euclid(A,B)
=> k9_metric_1(k14_euclid(np__2),D,C) = k6_jgraph_6(A,B,C) ) ) ) ) ) ).
fof(t49_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ( D = k23_euclid(A,B)
=> k11_metric_1(k14_euclid(np__2),D,C) = k5_jgraph_6(A,B,C) ) ) ) ) ) ).
fof(t50_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k1_topreal9(np__2,k23_euclid(A,B),C) = k6_jgraph_6(A,B,C) ) ) ) ).
fof(t51_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_topreal9(np__2,k23_euclid(A,B),C) = k7_jgraph_6(A,B,C) ) ) ) ).
fof(t52_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_topreal9(np__2,k23_euclid(A,B),C) = k5_jgraph_6(A,B,C) ) ) ) ).
fof(t53_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> r1_tarski(k6_jgraph_6(A,B,C),k7_jgraph_6(A,B,C)) ) ) ) ).
fof(t54_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> r1_xboole_0(k6_jgraph_6(A,B,C),k5_jgraph_6(A,B,C)) ) ) ) ).
fof(t55_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k4_subset_1(u1_struct_0(k15_euclid(np__2)),k6_jgraph_6(A,B,C),k5_jgraph_6(A,B,C)) = k7_jgraph_6(A,B,C) ) ) ) ).
fof(t56_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(B,k3_topreal9(np__2,k16_euclid(np__2),A))
=> k3_real_1(k7_square_1(k21_euclid(B)),k7_square_1(k22_euclid(B))) = k5_square_1(A) ) ) ) ).
fof(t57_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( D != E
& r2_hidden(D,k7_jgraph_6(A,B,C))
& r2_hidden(E,k7_jgraph_6(A,B,C))
& r1_xreal_0(C,np__0) ) ) ) ) ) ) ).
fof(t58_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(np__2))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(np__2))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(np__2))
=> ~ ( G = E
& H = F
& I = k23_euclid(A,B)
& C = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k3_xcmplx_0(np__2,k2_euclid_2(np__2,k20_euclid(np__2,F,E),k20_euclid(np__2,E,k23_euclid(A,B))))),k8_square_1(k1_quin_1(k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,H,G))),k3_xcmplx_0(np__2,k2_euclid_2(np__2,k20_euclid(np__2,F,E),k20_euclid(np__2,E,k23_euclid(A,B)))),k6_xcmplx_0(k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,G,I))),k5_square_1(D))))),k4_real_1(np__2,k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,H,G)))))
& E != F
& r2_hidden(E,k6_jgraph_6(A,B,D))
& ! [J] :
( m1_subset_1(J,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k1_struct_0(k15_euclid(np__2),J) = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal9(np__2,E,F),k5_jgraph_6(A,B,D))
& J = k17_euclid(np__2,k18_euclid(k6_xcmplx_0(np__1,C),np__2,E),k18_euclid(C,np__2,F)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t59_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,k5_jgraph_6(A,B,C))
& r2_hidden(E,k6_jgraph_6(A,B,C)) )
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,D,E),k5_jgraph_6(A,B,C)) = k1_struct_0(k15_euclid(np__2),D) ) ) ) ) ) ) ).
fof(t60_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,k5_jgraph_6(A,B,C))
& r2_hidden(E,k5_jgraph_6(A,B,C)) )
=> r1_tarski(k6_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,D,E),k2_struct_0(k15_euclid(np__2),D,E)),k6_jgraph_6(A,B,C)) ) ) ) ) ) ) ).
fof(t61_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,k5_jgraph_6(A,B,C))
& r2_hidden(E,k5_jgraph_6(A,B,C)) )
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,D,E),k5_jgraph_6(A,B,C)) = k2_struct_0(k15_euclid(np__2),D,E) ) ) ) ) ) ) ).
fof(t62_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,k5_jgraph_6(A,B,C))
& r2_hidden(E,k5_jgraph_6(A,B,C)) )
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal9(np__2,D,E),k5_jgraph_6(A,B,C)) = k2_struct_0(k15_euclid(np__2),D,E) ) ) ) ) ) ) ).
fof(t63_topreal9,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(np__2))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(np__2))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(np__2))
=> ~ ( G = k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F))
& H = F
& I = k23_euclid(A,B)
& E != F
& r2_hidden(E,k5_jgraph_6(A,B,C))
& r2_hidden(F,k7_jgraph_6(A,B,C))
& ! [J] :
( m1_subset_1(J,u1_struct_0(k15_euclid(np__2)))
=> ~ ( J != E
& k2_struct_0(k15_euclid(np__2),E,J) = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal9(np__2,E,F),k5_jgraph_6(A,B,C))
& ( r2_hidden(F,k3_topreal9(np__2,k23_euclid(A,B),C))
=> J = F )
& ( D = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k3_xcmplx_0(np__2,k2_euclid_2(np__2,k20_euclid(np__2,F,k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F))),k20_euclid(np__2,k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F)),k23_euclid(A,B))))),k8_square_1(k1_quin_1(k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,H,G))),k3_xcmplx_0(np__2,k2_euclid_2(np__2,k20_euclid(np__2,F,k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F))),k20_euclid(np__2,k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F)),k23_euclid(A,B)))),k6_xcmplx_0(k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,G,I))),k5_square_1(C))))),k4_real_1(np__2,k15_rvsum_1(k11_euclid(np__2,k8_euclid(np__2,H,G)))))
=> ( r2_hidden(F,k3_topreal9(np__2,k23_euclid(A,B),C))
| J = k17_euclid(np__2,k18_euclid(k6_xcmplx_0(np__1,D),np__2,k17_euclid(np__2,k18_euclid(k6_real_1(np__1,np__2),np__2,E),k18_euclid(k6_real_1(np__1,np__2),np__2,F))),k18_euclid(D,np__2,F)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> m1_subset_1(k1_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k2_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> m1_subset_1(k2_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k3_topreal9,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> m1_subset_1(k3_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k4_topreal9,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(k4_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(d1_topreal9,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)
=> k1_topreal9(A,B,C) = a_3_0_topreal9(A,B,C) ) ) ) ).
fof(d2_topreal9,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)
=> k2_topreal9(A,B,C) = a_3_1_topreal9(A,B,C) ) ) ) ).
fof(d3_topreal9,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)
=> k3_topreal9(A,B,C) = a_3_2_topreal9(A,B,C) ) ) ) ).
fof(d6_topreal9,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)))
=> k4_topreal9(A,B,C) = a_3_3_topreal9(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_topreal9,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& v1_xreal_0(D) )
=> ( r2_hidden(A,a_3_0_topreal9(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& ~ r1_xreal_0(D,k5_toprns_1(B,k20_euclid(B,E,C))) ) ) ) ).
fof(fraenkel_a_3_1_topreal9,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& v1_xreal_0(D) )
=> ( r2_hidden(A,a_3_1_topreal9(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& r1_xreal_0(k5_toprns_1(B,k20_euclid(B,E,C)),D) ) ) ) ).
fof(fraenkel_a_3_2_topreal9,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& v1_xreal_0(D) )
=> ( r2_hidden(A,a_3_2_topreal9(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& k5_toprns_1(B,k20_euclid(B,E,C)) = D ) ) ) ).
fof(fraenkel_a_3_3_topreal9,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_3_topreal9(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))
& r1_xreal_0(np__0,E) ) ) ) ).
%------------------------------------------------------------------------------